Table of discrete Probability Density Functions (PDFs): their properties and uses

 Name Origin/Use Form Parameters Properties Binomial Sequences of processes with two outcomes. Model Bernoulli trials. NCnpnqN-n p=success probability N=total trials n=number of successes (RV) E(n)=Np V(n)=Npq Poisson Number of random occurences per interval. Used to model arrivals in queueing. l ye-l /y! l =mean rate of success y=successes (RV) E(y)=l V(y)=l 2 Geometric Process of trial until success. qk-1p k=trials until success (RV) p=success probability E(k)=1/p V(k)=q/p2 Pascal or negative geometric Number of trials required for r successes k-1Cr-1qk-rpr k=trials required (RV) p=success probability E(k)=r/p V(k)=rq/p2 Hypergeometric A finite population of N things has r of one type and N-r of another. This distribution is the probability that a sample taken of n things has k of the said type in it. Very useful in deciding to accept/reject sampled lots. rCk*N-rCn-k ==========NCn k=success in sample (RV) N=total in lot n=sample size r=number of successes in lot   Approximate the hypergeometric distribution by the binomial well if n/N<0.1 E(k)=np V(k) = npq(N-n)======= (N-1) Multinomial A joint probability distribution. Like binomial, but for more than two possible outcomes. N!paapbb...pnn==========a!*b!*...*n! N=a+b+...+n N=total trials pa,pb,...,pn=probabilities of events a,b...,n a, b, ..., n=numbers of each type of event Not applicable This is a joint distribution. Empirical Observation p(x)=nx/N N=Total trials nx = number of x results

Note: V(k) means variance of RV k. E(k) means the expected value of k. NCn is a binomial coefficient; the number of combinations of N things taken n at a time.

Table of continuous Probability Density Functions; their properties and uses

 Name Origin/Use Form Parameters Properties Uniform Useful for modeling A/D errors, timing jitter, digitization error. 1/h for a

dof = degrees of freedom. E(y)=expected value of y. V(y)=variance of y=E(y2)-[E(y)]2

The normal distribution is placed in standard form with the transformation Z=(y-m )/s

Central Limit Theorem: If a sufficiently large sample be drawn forom any distribution that is unimodal and has tails that diminish rapidly, then the distribution of the mean of these samples will approximate a normal deviate t=(Y-E(y))*√(n/V(y)) where Y=sample mean, E(y)=expected value of the parent distribution (its mean), V(y)=variance of the parent distribution, and n=number of items in the sample.