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
==========
N
Cn

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<x<b

h=b-a

E(x)=(a+b)/2

V(x)=h≤/12

Gaussian or Normal

Continuous limit of binomial. Target pdf of Central Limit Theorem

Standard form.

(s v2p )-1e-(y-m )≤/2s

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

Gamma

Sum of independent exponential RVs.

a pyp-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 ≤=å Z≤i for i=1 to n

pdf is complex the table usually supplied is the cdf

Zi = independent standard normal RVs

n=number of Zs, degrees of freedom

E(Z)=n

V(Z)=2n.

Studentís 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

Snedecorís F or Fisher

Ratio of Chi squared RVs. Tests of difference of variance.

F=n2X≤1/n1X≤2

pdf is complex the table usually supplied is the cdf

X≤1 = Chisquare RV with n1 dof

X≤2 = 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.