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

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.	_{NC_npⁿq^N-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.	q^k-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-1C_r-1q^k-rp^r}	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.	_rC_k*_N-rC_{n-k ========== 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(N-n) ======= (N-1)
Multinomial	A joint probability distribution. Like binomial, but for more than two possible outcomes.	N!p_a^ap_b^b...p_nⁿ ========== a!b!...*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. _NC_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=b-a	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 )^-1e^-(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^-yb	b =expected failure rate y=time to fail (RV)	E(y)=b V(y)=b ²
Gamma	Sum of independent exponential RVs.	a ^py^p-1e^-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_ifor 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 (y-m )/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=n₂X²₁/n₁X²₂ pdf is complex the table usually supplied is the cdf	X²₁= Chisquare RV with n₁dof X²₂= Chisquare RV with n₂ dof	Not Applicable

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.