Useful Discrete Random Variable

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Useful Discrete Random Variable The University of Texas at Dallas Useful Discrete Random Variable Duncan MacFarlane The University of Texas at Dallas

Distributions and their uses Bernoulli – Number of successes on one trial Geometric – Number of trials until first success Binomial – Number of successes on n trials Pascal – Number of trials until the k-th success Poisson – Arrival statistics

Bernoulli and Geometric Distributions Number of successes on one trial P(x) = (1-p) for x=0, or P(x) = p for x=1 Geometric Number of trials until first success P(x) = p(1-p)(n-1)

Geometric PMF

Geometric CDF

Binomial Distribution Binomial – Number of Successes on n trials

Binomial PMF, n=10

Binomial CDF n=10

Pascal Distribution Pascal – Number of trials until the k-th success For n=k, k+1, k+2 ….

Arrival Statistics (busses, customers, etc) Poisson Distribution Arrival Statistics (busses, customers, etc) A: is the average arrival rate times the observation time A=T

Poisson PMF

Poisson CDF

Distribution E(X) Var(X) Summary Distribution E(X) Var(X) Bernoulli p p(1-p) Geometric 1/p (1-p)/p2 Binomial np np(1-p) Pascal k/p p(1-p)/p2 Poisson A