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. 
_{NCnpnqNn} 
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 ^{y}e^{}l /y! 
l =mean rate of success y=successes (RV) 
E(y)=l V(y)=l ^{2} 
Geometric 
Process of trial until success. 
q^{k1}p 
k=trials until success (RV) p=success probability 
E(k)=1/p V(k)=q/p^{2} 
Pascal or negative geometric 
Number of trials required for r successes 
_{k1Cr1qkrpr} 
k=trials required (RV) p=success probability 
E(k)=r/p V(k)=rq/p^{2} 
Hypergeometric 
A finite population of N things has r of one type and Nr 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. 
_{r}C_{k}*_{Nr}C_{nk ==========N}C_{n} 
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(Nn) 
Multinomial 
A joint probability distribution. Like binomial, but for more than two possible outcomes. 
N!p_{a}^{a}p_{b}^{b}...p_{n}^{n} N=a+b+...+n 
N=total trials p_{a,}p_{b,}...,p_{n}=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)=n_{x}/N 
N=Total trials n_{x }= number of x results 

Note: V(k) means variance of RV k. E(k) means the expected value of k. _{N}C_{n} 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<x<b 
h=ba 
E(x)=(a+b)/2 V(x)=h²/12 
Gaussian or Normal 
Continuous limit of binomial. Target pdf of Central Limit Theorem 
Standard form. (s v2p )^{1}e^{(y}m )²/2s ² 
s =standard dev. m =mean y=RV 
E(y)= m V(y)= s ^{2} 
Exponential 
Continuous limit of geometric. To model time to failure. 
b e^{y}b 
b =expected failure rate y=time to fail (RV) 
E(y)=b V(y)=b ² 
Gamma 
Sum of independent exponential RVs. 
a ^{p}y^{p1}e^{}a y/G (p) 
P=number of exponential RVs in sum. a =failure rate y=time to fail RV 
E(y)=p V(y)=p 
ChiSquared 
Sum of squared normal RVs. Tests of significance and goodness of fit. 
c ²=å Z²_{i }for i=1 to n pdf is complex the table usually supplied is the cdf_{ } 
Z_{i }= independent standard normal RVs n=number of Zs, degrees of freedom 
E(Z)=n V(Z)=2n. 
Student’s t 
Ratio of normal RV to square root of a Chi squared RV. Tests of difference of means. 
t=Ö n (ym )/s pdf is complex the table usually supplied is the cdf 
y=sample mean m =population mean s=sample std. dev. n=sample size 
Not Applicable 
Snedecor’s F or Fisher 
Ratio of Chi squared RVs. Tests of difference of variance. 
F=n_{2}X²_{1}/n_{1}X²_{2 } pdf is complex the table usually supplied is the cdf 
X²_{1 }= Chisquare RV with n_{1 }dof X²_{2 }= Chisquare RV with n_{2} dof 
Not Applicable 
dof = degrees of freedom. E(y)=expected value of y. V(y)=variance of y=E(y^{2})[E(y)]^{2}
The normal distribution is placed in standard form with the transformation Z=(ym )/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=(YE(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.