Useful Discrete Random Variables

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Presentation transcript:

Useful Discrete Random Variables Duncan MacFarlane The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane

PMF of Geometric Random Variables P(x) = px(1-p)(x-1) The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane

CDF of the Geometric Distribution P(x) = px(1-p)(x-1) The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane

PMF of Binomial Random Variable, n=10 (nx)px(1-p) (n-x) The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane

CDF of the Binomial Distribution (nx)px(1-p) (n-x) The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane

Poisson Random Variable -- arrival statistics P(x) = αxexp(-α)/x! α=T =(ave arriv/sec)(observ time) The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane

CDF of the Poisson Distribution αxexp(-α)/x! The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane

The Erik Jonsson School of Engineering and Computer Science Summary of Results Distribution Formula E[X] Var[X] Bernoulli (1-p) if x=0 Or p if x=1 p p(1-p) geometric px(1-p)(x-1) 1/p (1-p)/p2 binomial (nx)px(1-p) (n-x) np np(1-p) Pascal (x-1k-1)pk(1-p) (x-k) k/p p(1-p)/ p2 Poisson αxexp(-α)/x! α discrete uniform 1/(l-k-1) if x=k…l Or otherwise (k+l)/2 (l-k)(l-k+2)/12 The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane