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Mar 12, 2023 9 likes •141 views

Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE361: Signals and System II Probability Distributions http://www.ee.unlv.edu/~b1morris/ee361/ 2 Big Idea: Probability Distribution Assign a probability to each of the possible

SLIDE 1

http://www.ee.unlv.edu/~b1morris/ee361/ Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu

EE361: Signals and System II

Probability Distributions

SLIDE 2

Big Idea: Probability Distribution

Assign a probability to each of the possible
utcomes of a random experiment
Discrete

▫ Probability mass function (pmf) – probability of each possible outcome ▫ E.g. probability a roll of die will come up with a 3

Continuous

▫ Probability density function (pdf) – probability the outcome is within a range of values (interval) ▫ E.g. probability that a 500 g package is between 490-510 g

SLIDE 3

Special Distributions

Discrete

▫ Bernoulli ▫ Binomial ▫ Geometric ▫ Negative Binomial ▫ Poisson ▫ Uniform

Continuous

▫ Uniform ▫ Exponential ▫ Gamma ▫ Normal

SLIDE 4

Bernoulli Distribution

Binary RV with probability 𝑞 of 1 (“success”)

▫ E.g. a coin flip with heads a “success” or “1” and tails a “failure” or “0”

𝑞𝑌 𝑙 = 𝑄 𝑌 = 𝑙 = 𝑞𝑙 1 − 𝑞 1−𝑙

▫ 0 < 𝑞 < 1 is probability of success ▫ (1 − 𝑞) is probability of failure

4 pmf cdf

SLIDE 5

Binomial Distribution

RV to count the number of successes with 𝑜

independent Bernoulli trials

𝑞𝑌 𝑙 = 𝑄 𝑌 = 𝑙 = 𝑜

𝑙 𝑞𝑙 1 − 𝑞 𝑜−𝑙

▫ 𝑜 𝑙 - 𝑜 choose 𝑙 ways to get 𝑙 successes

5 pmf cdf

SLIDE 6

Geometric Distribution

Sequence of Bernoulli trials observed until first

success

𝑞𝑌 𝑦 = 𝑄 𝑌 = 𝑦 = 1 − 𝑞 𝑦−1𝑞

6 pmf cdf

SLIDE 7

Negative Binomial Distribution

Number of trials until 𝑙th success in sequence of

Bernoulli trials

𝑞𝑌 𝑦 = 𝑄 𝑌 = 𝑦 = 𝑦 − 1

𝑙 − 1 𝑞𝑙 1 − 𝑞 𝑦−𝑙

7 Note: parameter 𝑠 = 𝑦 − 𝑙 pmf

SLIDE 8

Poisson Distribution

The number of events occurring in a fixed

interval (time or space) given a known event average rate 𝜇

𝑞𝑌 𝑙 = 𝑄 𝑌 = 𝑙 = 𝑓−𝜇 𝜇𝑙

𝑙!

8 pmf cdf

SLIDE 9

Discrete Uniform Distribution

𝑞𝑌 𝑦 = 𝑄 𝑌 = 𝑦 =

1 𝑜

9 pmf cdf

SLIDE 10

Continuous Uniform Distribution

𝑔

𝑌 𝑦 = 1 𝑐−𝑏

𝑏 < 𝑦 < 𝑐 𝑓𝑚𝑡𝑓

10 pdf cdf

SLIDE 11

Exponential Distribution

𝑔

𝑌 𝑦 = 𝜇𝑓−𝜇𝑦 𝑦 > 0

11 pdf cdf

SLIDE 12

Gamma Distribution

𝑔

𝑌 𝑦 = 𝜇𝑓−𝜇𝑦 𝜇𝑦 𝛽−1 Γ 𝛽

𝑦 > 0

12 pdf cdf 𝛽 – shape parameter 𝜄 =

1 𝜇 rate parameter (inverse scale)

SLIDE 13

Normal (Gaussian) Distribution

𝑔

𝑌 𝑦 = 1 2𝜌𝜏2 𝑓− 𝑦−𝜈 2/2𝜏2

13 pdf cdf