EE361: Signals and System II Probability Distributions - PowerPoint PPT Presentation

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 outcomes 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

3 Special Distributions • Discrete • Continuous ▫ Bernoulli ▫ Uniform ▫ Binomial ▫ Exponential ▫ Geometric ▫ Gamma ▫ Negative Binomial ▫ Normal ▫ Poisson ▫ Uniform

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” • 𝑞 𝑌 𝑙 = 𝑄 𝑌 = 𝑙 = 𝑞 𝑙 1 − 𝑞 1−𝑙 ▫ 0 < 𝑞 < 1 is probability of success ▫ (1 − 𝑞) is probability of failure pmf cdf

5 Binomial Distribution • RV to count the number of successes with 𝑜 independent Bernoulli trials • 𝑞 𝑌 𝑙 = 𝑄 𝑌 = 𝑙 = 𝑜 𝑙 𝑞 𝑙 1 − 𝑞 𝑜−𝑙 𝑜 𝑙 - 𝑜 choose 𝑙 ways to get 𝑙 successes ▫ pmf cdf

6 Geometric Distribution • Sequence of Bernoulli trials observed until first success • 𝑞 𝑌 𝑦 = 𝑄 𝑌 = 𝑦 = 1 − 𝑞 𝑦−1 𝑞 pmf cdf

7 Negative Binomial Distribution • Number of trials until 𝑙 th success in sequence of Bernoulli trials • 𝑞 𝑌 𝑦 = 𝑄 𝑌 = 𝑦 = 𝑦 − 1 𝑙 − 1 𝑞 𝑙 1 − 𝑞 𝑦−𝑙 Note: parameter 𝑠 = 𝑦 − 𝑙 pmf

8 Poisson Distribution • The number of events occurring in a fixed interval (time or space) given a known event average rate 𝜇 • 𝑞 𝑌 𝑙 = 𝑄 𝑌 = 𝑙 = 𝑓 −𝜇 𝜇 𝑙 𝑙! pmf cdf

9 Discrete Uniform Distribution 1 • 𝑞 𝑌 𝑦 = 𝑄 𝑌 = 𝑦 = 𝑜 pmf cdf

10 Continuous Uniform Distribution 1 𝑏 < 𝑦 < 𝑐 • 𝑔 𝑌 𝑦 = 𝑐−𝑏 0 𝑓𝑚𝑡𝑓 pdf cdf

11 Exponential Distribution 𝑌 𝑦 = 𝜇𝑓 −𝜇𝑦 𝑦 > 0 • 𝑔 pdf cdf

12 Gamma Distribution 𝜇𝑓 −𝜇𝑦 𝜇𝑦 𝛽−1 • 𝑔 𝑌 𝑦 = 𝑦 > 0 Γ 𝛽 pdf cdf 𝛽 – shape parameter 1 𝜄 = 𝜇 rate parameter (inverse scale)

13 Normal (Gaussian) Distribution 1 2𝜌𝜏 2 𝑓 − 𝑦−𝜈 2 /2𝜏 2 • 𝑔 𝑌 𝑦 = pdf cdf