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logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

The Poisson Distribution

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Introduction

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Introduction

Some counting processes occur over time or they rely on another continuous parameter.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Introduction

Some counting processes occur over time or they rely on another continuous parameter.

• 1. The number of times a given web page is hit in a certain

period of time is a random variable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Introduction

Some counting processes occur over time or they rely on another continuous parameter.

• 1. The number of times a given web page is hit in a certain

period of time is a random variable.

• 2. The number of calls to customer support of a company in a

certain period of time is a random variable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Introduction

Some counting processes occur over time or they rely on another continuous parameter.

• 1. The number of times a given web page is hit in a certain

period of time is a random variable.

• 2. The number of calls to customer support of a company in a

certain period of time is a random variable.

• 3. The number of radioactive particles registered in a Geiger

counter over a certain period of time is a random variable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Introduction

Some counting processes occur over time or they rely on another continuous parameter.

• 1. The number of times a given web page is hit in a certain

period of time is a random variable.

• 2. The number of calls to customer support of a company in a

certain period of time is a random variable.

• 3. The number of radioactive particles registered in a Geiger

counter over a certain period of time is a random variable. But what’s the sample space and how do we assign probabilities?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Introduction

Some counting processes occur over time or they rely on another continuous parameter.

• 1. The number of times a given web page is hit in a certain

period of time is a random variable.

• 2. The number of calls to customer support of a company in a

certain period of time is a random variable.

• 3. The number of radioactive particles registered in a Geiger

counter over a certain period of time is a random variable. But what’s the sample space and how do we assign probabilities? We’ll actually sidestep the sample space issue

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Introduction

Some counting processes occur over time or they rely on another continuous parameter.

• 1. The number of times a given web page is hit in a certain

period of time is a random variable.

• 2. The number of calls to customer support of a company in a

certain period of time is a random variable.

• 3. The number of radioactive particles registered in a Geiger

counter over a certain period of time is a random variable. But what’s the sample space and how do we assign probabilities? We’ll actually sidestep the sample space issue (it’s complicated)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Introduction

Some counting processes occur over time or they rely on another continuous parameter.

• 1. The number of times a given web page is hit in a certain

period of time is a random variable.

• 2. The number of calls to customer support of a company in a

certain period of time is a random variable.

• 3. The number of radioactive particles registered in a Geiger

counter over a certain period of time is a random variable. But what’s the sample space and how do we assign probabilities? We’ll actually sidestep the sample space issue (it’s complicated) and we’ll focus on the probabilities.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Introduction

Some counting processes occur over time or they rely on another continuous parameter.

• 1. The number of times a given web page is hit in a certain

period of time is a random variable.

• 2. The number of calls to customer support of a company in a

certain period of time is a random variable.

• 3. The number of radioactive particles registered in a Geiger

counter over a certain period of time is a random variable. But what’s the sample space and how do we assign probabilities? We’ll actually sidestep the sample space issue (it’s complicated) and we’ll focus on the probabilities. For visualization, we assume the process occurs over time.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4 x5

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4 x5 x6

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4 x5 x6 ···

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4 x5 x6 ··· xn−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4 x5 x6 ··· xn−1

That means there is a λ > 0 so that the probability that an event

• ccurs in a given time interval is p = λ

n .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4 x5 x6 ··· xn−1

That means there is a λ > 0 so that the probability that an event

• ccurs in a given time interval is p = λ

n . We keep this λ independent of the number of subintervals.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4 x5 x6 ··· xn−1

That means there is a λ > 0 so that the probability that an event

• ccurs in a given time interval is p = λ

n . We keep this λ independent of the number of subintervals. That way, if we double the number of intervals, for each interval the probability

• f an event occurring is divided by 2.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4 x5 x6 ··· xn−1

That means there is a λ > 0 so that the probability that an event

• ccurs in a given time interval is p = λ

n . We keep this λ independent of the number of subintervals. That way, if we double the number of intervals, for each interval the probability

• f an event occurring is divided by 2.

These assumptions are realistic

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4 x5 x6 ··· xn−1

That means there is a λ > 0 so that the probability that an event

• ccurs in a given time interval is p = λ

n . We keep this λ independent of the number of subintervals. That way, if we double the number of intervals, for each interval the probability

• f an event occurring is divided by 2.

These assumptions are realistic for hits on web pages

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4 x5 x6 ··· xn−1

That means there is a λ > 0 so that the probability that an event

• ccurs in a given time interval is p = λ

n . We keep this λ independent of the number of subintervals. That way, if we double the number of intervals, for each interval the probability

• f an event occurring is divided by 2.

These assumptions are realistic for hits on web pages, for customer support calls received

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals.

✲ s s s s s s x0 = 0 xn = T x1 x2 x3 x4 x5 x6 ··· xn−1

That means there is a λ > 0 so that the probability that an event

• ccurs in a given time interval is p = λ

n . We keep this λ independent of the number of subintervals. That way, if we double the number of intervals, for each interval the probability

• f an event occurring is divided by 2.

These assumptions are realistic for hits on web pages, for customer support calls received, for radioactive particles registered.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s x0 = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s x0 = 0 x1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s x0 = 0 x1 x2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s x0 = 0 x1 x2 x3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s x0 = 0 x1 x2 x3 x4

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s x0 = 0 x1 x2 x3 x4 x5

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s x0 = 0 x1 x2 x3 x4 x5 x6

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s x0 = 0 x1 x2 x3 x4 x5 x6 ···

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s x0 = 0 x1 x2 x3 x4 x5 x6 ··· xn−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s x0 = 0 x1 x2 x3 x4 x5 x6 ··· xn−1 xn = T

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n) this is realistic in the above examples.

✲ s s s s s s x0 = 0 x1 x2 x3 x4 x5 x6 ··· xn−1 xn = T

Plus, we could argue that within time intervals of a certain length only one event can be registered.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval,

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. b(x;n,p)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. b(x;n,p) = n x

• px(1−p)n−x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. b(x;n,p) = n x

• px(1−p)n−x =

n! (n−x)!x!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. b(x;n,p) = n x

• px(1−p)n−x =

n! (n−x)!x! λ x nx

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. b(x;n,p) = n x

• px(1−p)n−x =

n! (n−x)!x! λ x nx

• 1− λ

n n−x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. b(x;n,p) = n x

• px(1−p)n−x =

n! (n−x)!x! λ x nx

• 1− λ

n n−x = n! (n−x)!nx λ x x!

• 1− λ

n n n−x

n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. b(x;n,p) = n x

• px(1−p)n−x =

n! (n−x)!x! λ x nx

• 1− λ

n n−x = n! (n−x)!nx λ x x!

• 1− λ

n n n−x

n

n→∞

− → 1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. b(x;n,p) = n x

• px(1−p)n−x =

n! (n−x)!x! λ x nx

• 1− λ

n n−x = n! (n−x)!nx λ x x!

• 1− λ

n n n−x

n

n→∞

− → 1· λ x x!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. b(x;n,p) = n x

• px(1−p)n−x =

n! (n−x)!x! λ x nx

• 1− λ

n n−x = n! (n−x)!nx λ x x!

• 1− λ

n n n−x

n

n→∞

− → 1· λ x x! ·e−λ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. b(x;n,p) = n x

• px(1−p)n−x =

n! (n−x)!x! λ x nx

• 1− λ

n n−x = n! (n−x)!nx λ x x!

• 1− λ

n n n−x

n

n→∞

− → 1· λ x x! ·e−λ (We let n → ∞ because cutting up the original time interval was just a way to get the model.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Definition. A random variable is said to have a Poisson

distribution with parameter λ > 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Definition. A random variable is said to have a Poisson

distribution with parameter λ > 0 if and only if its probability mass function is p(x;λ) = e−λλ x x!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Definition. A random variable is said to have a Poisson

distribution with parameter λ > 0 if and only if its probability mass function is p(x;λ) = e−λλ x x! This really is a probability mass function, because of the series representation of the exponential function

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Definition. A random variable is said to have a Poisson

distribution with parameter λ > 0 if and only if its probability mass function is p(x;λ) = e−λλ x x! This really is a probability mass function, because of the series representation of the exponential function: eλ =

∞

∑

x=0

λ x x! .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. The number of calls a certain customer support

department receives in any given 5 minute interval is Poisson distributed with λ = 2.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. The number of calls a certain customer support

department receives in any given 5 minute interval is Poisson distributed with λ = 2. What is the probability that at most 3 calls are received in the next 5 minutes?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. The number of calls a certain customer support

department receives in any given 5 minute interval is Poisson distributed with λ = 2. What is the probability that at most 3 calls are received in the next 5 minutes? p(0;2)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. The number of calls a certain customer support

department receives in any given 5 minute interval is Poisson distributed with λ = 2. What is the probability that at most 3 calls are received in the next 5 minutes? p(0;2)+p(1;2)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. The number of calls a certain customer support

department receives in any given 5 minute interval is Poisson distributed with λ = 2. What is the probability that at most 3 calls are received in the next 5 minutes? p(0;2)+p(1;2)+p(2;2)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. The number of calls a certain customer support

department receives in any given 5 minute interval is Poisson distributed with λ = 2. What is the probability that at most 3 calls are received in the next 5 minutes? p(0;2)+p(1;2)+p(2;2)+p(3;2)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. The number of calls a certain customer support

department receives in any given 5 minute interval is Poisson distributed with λ = 2. What is the probability that at most 3 calls are received in the next 5 minutes? p(0;2)+p(1;2)+p(2;2)+p(3;2) = 0.857

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. The number of calls a certain customer support

department receives in any given 5 minute interval is Poisson distributed with λ = 2. What is the probability that at most 3 calls are received in the next 5 minutes? p(0;2)+p(1;2)+p(2;2)+p(3;2) = 0.857 (Used a Poisson distribution table to look up p(x ≤ 3;2), which is just a value of the cumulative distribution function.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

For large values of n, the Poisson distribution can be used to approximate binomial probabilities.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

For large values of n, the Poisson distribution can be used to approximate binomial probabilities. Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

For large values of n, the Poisson distribution can be used to approximate binomial probabilities.

• Example. Suppose the probability of a missing page in an

individual book a certain manufacturer makes is 0.1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

For large values of n, the Poisson distribution can be used to approximate binomial probabilities.

• Example. Suppose the probability of a missing page in an

individual book a certain manufacturer makes is 0.1. What is the probability that a batch of 200 books has at most 10 books with a page missing?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

For large values of n, the Poisson distribution can be used to approximate binomial probabilities.

• Example. Suppose the probability of a missing page in an

individual book a certain manufacturer makes is 0.1. What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

For large values of n, the Poisson distribution can be used to approximate binomial probabilities.

• Example. Suppose the probability of a missing page in an

individual book a certain manufacturer makes is 0.1. What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that λ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

For large values of n, the Poisson distribution can be used to approximate binomial probabilities.

• Example. Suppose the probability of a missing page in an

individual book a certain manufacturer makes is 0.1. What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that λ = np

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

For large values of n, the Poisson distribution can be used to approximate binomial probabilities.

• Example. Suppose the probability of a missing page in an

individual book a certain manufacturer makes is 0.1. What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that λ = np = 200·0.1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

For large values of n, the Poisson distribution can be used to approximate binomial probabilities.

• Example. Suppose the probability of a missing page in an

individual book a certain manufacturer makes is 0.1. What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that λ = np = 200·0.1 = 20.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

For large values of n, the Poisson distribution can be used to approximate binomial probabilities.

• Example. Suppose the probability of a missing page in an

individual book a certain manufacturer makes is 0.1. What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that λ = np = 200·0.1 = 20. Now from a Poisson distribution table, p(x ≤ 10;20) ≈ 0.011.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

For large values of n, the Poisson distribution can be used to approximate binomial probabilities.

• Example. Suppose the probability of a missing page in an

individual book a certain manufacturer makes is 0.1. What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that λ = np = 200·0.1 = 20. Now from a Poisson distribution table, p(x ≤ 10;20) ≈ 0.011. With a CAS we obtain B(10;200,0.1) ≈ 0.008.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Theorem. Let X be a Poisson distributed random variable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Theorem. Let X be a Poisson distributed random variable.

Then E(X) = λ and V(X) = λ.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Theorem. Let X be a Poisson distributed random variable.

Then E(X) = λ and V(X) = λ. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Theorem. Let X be a Poisson distributed random variable.

Then E(X) = λ and V(X) = λ. Proof. E(X)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Theorem. Let X be a Poisson distributed random variable.

Then E(X) = λ and V(X) = λ. Proof. E(X) =

∞

∑

x=0

xe−λλ x x!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Theorem. Let X be a Poisson distributed random variable.

Then E(X) = λ and V(X) = λ. Proof. E(X) =

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=1

e−λλ x (x−1)!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Theorem. Let X be a Poisson distributed random variable.

Then E(X) = λ and V(X) = λ. Proof. E(X) =

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=1

e−λλ x (x−1)! = λe−λ

∞

∑

x=1

λ x−1 (x−1)!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Theorem. Let X be a Poisson distributed random variable.

Then E(X) = λ and V(X) = λ. Proof. E(X) =

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=1

e−λλ x (x−1)! = λe−λ

∞

∑

x=1

λ x−1 (x−1)! = λe−λ

∞

∑

j=0

λ j j!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Theorem. Let X be a Poisson distributed random variable.

Then E(X) = λ and V(X) = λ. Proof. E(X) =

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=1

e−λλ x (x−1)! = λe−λ

∞

∑

x=1

λ x−1 (x−1)! = λe−λ

∞

∑

j=0

λ j j! = λe−λeλ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Theorem. Let X be a Poisson distributed random variable.

Then E(X) = λ and V(X) = λ. Proof. E(X) =

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=1

e−λλ x (x−1)! = λe−λ

∞

∑

x=1

λ x−1 (x−1)! = λe−λ

∞

∑

j=0

λ j j! = λe−λeλ = λ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x! =

∞

∑

x=0

x(x−1)e−λλ x x! +

∞

∑

x=0

xe−λλ x x!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x! =

∞

∑

x=0

x(x−1)e−λλ x x! +

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=2

x(x−1)e−λλ x x! +λ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x! =

∞

∑

x=0

x(x−1)e−λλ x x! +

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=2

x(x−1)e−λλ x x! +λ = e−λλ 2

∞

∑

x=2

λ x−2 (x−2)! +λ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x! =

∞

∑

x=0

x(x−1)e−λλ x x! +

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=2

x(x−1)e−λλ x x! +λ = e−λλ 2

∞

∑

x=2

λ x−2 (x−2)! +λ = e−λλ 2

∞

∑

k=0

λ k k! +λ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x! =

∞

∑

x=0

x(x−1)e−λλ x x! +

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=2

x(x−1)e−λλ x x! +λ = e−λλ 2

∞

∑

x=2

λ x−2 (x−2)! +λ = e−λλ 2

∞

∑

k=0

λ k k! +λ = e−λλ 2eλ +λ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x! =

∞

∑

x=0

x(x−1)e−λλ x x! +

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=2

x(x−1)e−λλ x x! +λ = e−λλ 2

∞

∑

x=2

λ x−2 (x−2)! +λ = e−λλ 2

∞

∑

k=0

λ k k! +λ = e−λλ 2eλ +λ = λ 2 +λ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x! =

∞

∑

x=0

x(x−1)e−λλ x x! +

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=2

x(x−1)e−λλ x x! +λ = e−λλ 2

∞

∑

x=2

λ x−2 (x−2)! +λ = e−λλ 2

∞

∑

k=0

λ k k! +λ = e−λλ 2eλ +λ = λ 2 +λ V(X)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x! =

∞

∑

x=0

x(x−1)e−λλ x x! +

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=2

x(x−1)e−λλ x x! +λ = e−λλ 2

∞

∑

x=2

λ x−2 (x−2)! +λ = e−λλ 2

∞

∑

k=0

λ k k! +λ = e−λλ 2eλ +λ = λ 2 +λ V(X) = E

• X2

−E(X)2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x! =

∞

∑

x=0

x(x−1)e−λλ x x! +

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=2

x(x−1)e−λλ x x! +λ = e−λλ 2

∞

∑

x=2

λ x−2 (x−2)! +λ = e−λλ 2

∞

∑

k=0

λ k k! +λ = e−λλ 2eλ +λ = λ 2 +λ V(X) = E

• X2

−E(X)2 = λ 2 +λ −λ 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x! =

∞

∑

x=0

x(x−1)e−λλ x x! +

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=2

x(x−1)e−λλ x x! +λ = e−λλ 2

∞

∑

x=2

λ x−2 (x−2)! +λ = e−λλ 2

∞

∑

k=0

λ k k! +λ = e−λλ 2eλ +λ = λ 2 +λ V(X) = E

• X2

−E(X)2 = λ 2 +λ −λ 2 = λ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

E

• X2

=

∞

∑

x=0

x2e−λλ x x! =

∞

∑

x=0

x(x−1)e−λλ x x! +

∞

∑

x=0

xe−λλ x x! =

∞

∑

x=2

x(x−1)e−λλ x x! +λ = e−λλ 2

∞

∑

x=2

λ x−2 (x−2)! +λ = e−λλ 2

∞

∑

k=0

λ k k! +λ = e−λλ 2eλ +λ = λ 2 +λ V(X) = E

• X2

−E(X)2 = λ 2 +λ −λ 2 = λ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Extending to Arbitrary Intervals

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Extending to Arbitrary Intervals

A Poisson process is a stochastic process with the following properties.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Extending to Arbitrary Intervals

A Poisson process is a stochastic process with the following properties.

• 1. There is an α > 0 so that for a short interval ∆t the

probability of one event is α∆t +o(∆t).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Extending to Arbitrary Intervals

A Poisson process is a stochastic process with the following properties.

• 1. There is an α > 0 so that for a short interval ∆t the

probability of one event is α∆t +o(∆t).

• 2. The probability of more than one event is o(∆t).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Extending to Arbitrary Intervals

A Poisson process is a stochastic process with the following properties.

• 1. There is an α > 0 so that for a short interval ∆t the

probability of one event is α∆t +o(∆t).

• 2. The probability of more than one event is o(∆t).
• 3. The number of events in an interval ∆t is independent of

the number received in other intervals.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Extending to Arbitrary Intervals

A Poisson process is a stochastic process with the following properties.

• 1. There is an α > 0 so that for a short interval ∆t the

probability of one event is α∆t +o(∆t).

• 2. The probability of more than one event is o(∆t).
• 3. The number of events in an interval ∆t is independent of

the number received in other intervals. In a Poisson process with parameter α, the probability of k events in an interval of length t is Poisson distributed with parameter λ = αt.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Extending to Arbitrary Intervals

A Poisson process is a stochastic process with the following properties.

• 1. There is an α > 0 so that for a short interval ∆t the

probability of one event is α∆t +o(∆t).

• 2. The probability of more than one event is o(∆t).
• 3. The number of events in an interval ∆t is independent of

the number received in other intervals. In a Poisson process with parameter α, the probability of k events in an interval of length t is Poisson distributed with parameter λ = αt. Pk(t) = e−αt (αt)k k! .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second. The probability of registering a particle in a time interval is proportional to the length ∆t of the interval

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second. The probability of registering a particle in a time interval is proportional to the length ∆t of the interval, so it’s α∆t.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second. The probability of registering a particle in a time interval is proportional to the length ∆t of the interval, so it’s α∆t. For really short time intervals, the probability of registering two particles is zero

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second. The probability of registering a particle in a time interval is proportional to the length ∆t of the interval, so it’s α∆t. For really short time intervals, the probability of registering two particles is zero (particles will arrive at different times

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second. The probability of registering a particle in a time interval is proportional to the length ∆t of the interval, so it’s α∆t. For really short time intervals, the probability of registering two particles is zero (particles will arrive at different times, counter must “reset”).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second. The probability of registering a particle in a time interval is proportional to the length ∆t of the interval, so it’s α∆t. For really short time intervals, the probability of registering two particles is zero (particles will arrive at different times, counter must “reset”). The number of particles received in a time interval does not depend on the past.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second. The probability of registering a particle in a time interval is proportional to the length ∆t of the interval, so it’s α∆t. For really short time intervals, the probability of registering two particles is zero (particles will arrive at different times, counter must “reset”). The number of particles received in a time interval does not depend on the past. So this is a Poisson process with α = 2 and interval length t = 1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second. The probability of registering a particle in a time interval is proportional to the length ∆t of the interval, so it’s α∆t. For really short time intervals, the probability of registering two particles is zero (particles will arrive at different times, counter must “reset”). The number of particles received in a time interval does not depend on the past. So this is a Poisson process with α = 2 and interval length t = 1. P(X ≥ 4)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second. The probability of registering a particle in a time interval is proportional to the length ∆t of the interval, so it’s α∆t. For really short time intervals, the probability of registering two particles is zero (particles will arrive at different times, counter must “reset”). The number of particles received in a time interval does not depend on the past. So this is a Poisson process with α = 2 and interval length t = 1. P(X ≥ 4) = 1−P0(1)−P1(1)−P2(1)−P3(1)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second. The probability of registering a particle in a time interval is proportional to the length ∆t of the interval, so it’s α∆t. For really short time intervals, the probability of registering two particles is zero (particles will arrive at different times, counter must “reset”). The number of particles received in a time interval does not depend on the past. So this is a Poisson process with α = 2 and interval length t = 1. P(X ≥ 4) = 1−P0(1)−P1(1)−P2(1)−P3(1) = 1−e−220 0! −e−221 1! −e−222 2! −e−223 3!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Particles are registered by a Geiger counter at an

average rate of 2 particles per second. Find the probability that 4 or more particles are registered in a given second. The probability of registering a particle in a time interval is proportional to the length ∆t of the interval, so it’s α∆t. For really short time intervals, the probability of registering two particles is zero (particles will arrive at different times, counter must “reset”). The number of particles received in a time interval does not depend on the past. So this is a Poisson process with α = 2 and interval length t = 1. P(X ≥ 4) = 1−P0(1)−P1(1)−P2(1)−P3(1) = 1−e−220 0! −e−221 1! −e−222 2! −e−223 3! ≈ 0.1429

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road, so it’s α∆s.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road, so it’s α∆s. For really short lengths ∆s, the probability of two infested trees is 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road, so it’s α∆s. For really short lengths ∆s, the probability of two infested trees is 0. (How many trees are 1 in apart?)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road, so it’s α∆s. For really short lengths ∆s, the probability of two infested trees is 0. (How many trees are 1 in apart?) The number of infested trees on any given quarter mile is assumed to be independent of the number on previous stretches of road.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road, so it’s α∆s. For really short lengths ∆s, the probability of two infested trees is 0. (How many trees are 1 in apart?) The number of infested trees on any given quarter mile is assumed to be independent of the number on previous stretches of road. So this is a Poisson process with α = 12, if we use miles as the unit

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road, so it’s α∆s. For really short lengths ∆s, the probability of two infested trees is 0. (How many trees are 1 in apart?) The number of infested trees on any given quarter mile is assumed to be independent of the number on previous stretches of road. So this is a Poisson process with α = 12, if we use miles as the unit, and the length is s = 1/2.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road, so it’s α∆s. For really short lengths ∆s, the probability of two infested trees is 0. (How many trees are 1 in apart?) The number of infested trees on any given quarter mile is assumed to be independent of the number on previous stretches of road. So this is a Poisson process with α = 12, if we use miles as the unit, and the length is s = 1/2. P(X = 0)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road, so it’s α∆s. For really short lengths ∆s, the probability of two infested trees is 0. (How many trees are 1 in apart?) The number of infested trees on any given quarter mile is assumed to be independent of the number on previous stretches of road. So this is a Poisson process with α = 12, if we use miles as the unit, and the length is s = 1/2. P(X = 0) = P0 1 2

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road, so it’s α∆s. For really short lengths ∆s, the probability of two infested trees is 0. (How many trees are 1 in apart?) The number of infested trees on any given quarter mile is assumed to be independent of the number on previous stretches of road. So this is a Poisson process with α = 12, if we use miles as the unit, and the length is s = 1/2. P(X = 0) = P0 1 2

• = e−12· 1

2

• 12· 1

2

0!

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road, so it’s α∆s. For really short lengths ∆s, the probability of two infested trees is 0. (How many trees are 1 in apart?) The number of infested trees on any given quarter mile is assumed to be independent of the number on previous stretches of road. So this is a Poisson process with α = 12, if we use miles as the unit, and the length is s = 1/2. P(X = 0) = P0 1 2

• = e−12· 1

2

• 12· 1

2

0! = e−6

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution

logo1 Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process

• Example. Driving along a road, trees are seen to be kudzu

infested at an average rate of 3 trees per quarter mile. Find the probability of encountering a kudzu free half mile stretch. The probability of seeing an infested tree over a stretch of road is proportional to the length ∆s of the stretch of road, so it’s α∆s. For really short lengths ∆s, the probability of two infested trees is 0. (How many trees are 1 in apart?) The number of infested trees on any given quarter mile is assumed to be independent of the number on previous stretches of road. So this is a Poisson process with α = 12, if we use miles as the unit, and the length is s = 1/2. P(X = 0) = P0 1 2

• = e−12· 1

2

• 12· 1

2

0! = e−6 ≈ 0.002479.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Poisson Distribution