Advanced Mathematical Methods Part II Statistics Probability - - PowerPoint PPT Presentation

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Advanced Mathematical Methods

Part II – Statistics Probability Distributions

Mel Slater

http://www.cs.ucl.ac.uk/staff/m.slater/Teaching/Statistics/ www.astrocosmo.cl/biografi/b-c_gauss.htm

Carl Friedrich Gauss

www.math.yorku.ca/SCS/Gallery/bright-ideas.html

Siméon Poisson

Two famous probability distributions Poisson and Normal (Gaussian)

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Outline

Random variables Probability Distributions Summary Measures Moment Generating Functions (Common Distributions) In today’s lecture we learn some

definitions that are critical for the rest of the course.

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Random Variables

Events are subsets of the domain of

elementary events

Very often the ‘event’ is expressed in

numerical form

• The number of people out of n who

experience high presence in a VE is x

• The number of heads on n tosses of a coin is

x

• The heart rate of a person when subject to a

certain stimulii will increase by x

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Random Variables

A random variable is a variable that

has a probability associated with each possible value it can take (discrete case)

Or which has a probability

associated with any subset of the range of values it can take.

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Random Variable Examples

For example: X is the number of heads in

two tosses of a coin, then

• p(X=0) = ¼

{TT}

• p(X=1) = ½

{HT,TH}

• p(X=2) = ¼

{HH}

Y is the number of cars that pass a spot

• n a road during 1 hour. The range of

values of X = 0,1,2,3,…

Z is the time between two successive

cars passing a spot on a road, then Y ≥0 (a continuous r.v.).

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Distribution Function

For any r.v. X, define:

• F(x) = p(X ≤ x)
• The probability that X will have a value of no

more than x.

Suppose the range of X is [a,b]. Then it

follows from the axioms of probability that:

• F(a) = 0
• F(b) = 1
• F(x+h) – F(x) ≥ 0 for any h ≥ 0.

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Discrete Distributions

When X is a discrete r.v. with integer

values then, define

• f(x) = F(x+1) – F(x)

Note f(x) = p(X=x)

• the probability that X takes the value x

exactly.

f is called the probability density

function (pdf).

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Suppose X is a continuous r.v., and consider f(x) is also the p.d.f. and may be interpreted as f(x)dx

being the probability that X takes a value in a small interval around x.

Continuous Distributions

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Distributions and Probability

Note that

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Properties of pdf’s

Any function that satisfies these properties may serve as a pdf.

where D is the range of X.

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Summary Measures

Expectation (Mean) In general if g is any function of X then It is easy to show from the definition that

• E(a + bg(X)) = a + bE(g(X)) for constants a,b.

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Variance

The variance is a measure of variation

about the mean:-

If we write

Then it can be shown that

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Moments about Origin & Mean

rth moment about the origin is defined as rth moment about the mean as:

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Mean and Variance as Moments

Show:- Note that some distributions will have no finite moments. The moments of a distribution (if they exist) completely

characterise the distribution.

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Different way to specify a distribution Expand the exponential to get: So the moments are coefficients of

powers of t over the factorials!

Moment Generating Functions

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MGF Properties

where g(X) is any function of X

(prove by differentiation)

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Distributions

There are many ‘standard’

distributions

These occur again and again in

scientific applications

And are given special names. We will consider several now.

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Discrete Uniform Distribution

X is a r.v. which has domain {1,2,…,n} Each value of X is considered equally

likely.

P(X=x) = 1/n, x = 1,2,…,n

• (there are many possible versions of this

distribution)

Ex: From the definition find E(X), Var(X)

and MX(t)

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Continuous Uniform Distribution

X is equally likely

to take any value in the continuous range [a,b]:

The MGF can be

found to be:

From this it is easy

to find the mean E(X) and variance Var(X).

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Binomial Distribution

There are two exclusive outcomes to

an experimental trial labelled S or F.

P(S) = p, P(F) = 1-p = q. Each trial is independent. There are n trials. X is the number of S outcomes. X ∈ {0,1,2,…,n}

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Binomial Distribution

Any particular outcome with x S’s has

probability pxqn-x.

Combinatorial theory shows that there are

n!/x!(n-x)! such outcomes with x S’s and n-x Fs.

Hence:- With MGF With E(X) = np and Var(X) = npq

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Poisson Distribution

An event happens at

random in time

• E.g., failures of the

tracking system in the ‘Cave’

• E.g., a customer

enters a shop

• E.g., a train arrives at

a station

X is the number of

events per unit time

X = 0,1,2,…

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Exponential Distribution

Same setup as for

Poisson but let Y be the time between successive events.

MGF is: Expanding the MGF

as a power series it is easy to see that:

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Other Distributions…

Check out each of the following in the

notes:

• Beta distribution

– Distribution proportions i.e. in [0,1]

• Negative binomial distribution

– The number of trials before x successes are achieved in the binomial setup

• Gamma Distribution

– This is the time between k successive events in the Poisson setup.

For each case look at the derivation,

mean, variance and MGF

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Standard Statistical Distributions

There are a number of distributions that

regularly appear in statistical practice:

• Normal (Gaussian)
• Chi-squared χ2
• t-distribution
• F distribution

With MATLAB you should learn how to

find probabilities from these distributions.

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Normal Distribution

Occurs very often in

nature and in social situations – arises from an ‘averaging’ process.

The mean E(X) = µ Var(X) = σ2

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Standard Normal

X ~ N(µ, σ2) When µ = 0 and σ2 = 1 this is called the Standard

Normal distribution.

It is easy to show that when

X ~ N(µ, σ2)

Then Z = (X- µ)/σ ~ N(0,1) Then eg,

• P(X > a) = P(Z > (a- µ)/σ )
• and Z is tabulated (or available in MATLAB)

Typically we use P(Z<z) = F(z) the cummulative

probability distribution (distribution function).

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Chi-Squared χ2 Distribution

Let X be a r.v. which is formed by

the sum of squares of n independent N(0,1) variables.

X ~ χ2(n) distribution where n called

the ‘degrees of freedom’.

Use MATLAB to learn how to look

up probabilities for this distribution.

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t-Distribution

Suppose Z ~ N(0,1) X ~ χ2(n) Suppose Z and X are independent. Then t = Z/(X/n) ~ t-distribution with

n degrees of freedom.

Learn how to use MATLAB to look

up probabilities for this distribution.

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F-Distribution

Suppose X1 ~ χ2(n1) Suppose X2 ~ χ2(n2)

• X1 and X2 are independent

Then

• F = (X1/n1) / (X2/n2) ~ F(n1,n2) distribution
• Where n1 and n2 are called the degrees of

freedom of numerator and denominator.

Learn how to look up probabilities under

this distribution using MATLAB.