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Discrete Random Variables: Summary of the Discrete Probability Functions

Module by: Dr. Barbara Illowsky, Susan Dean

Summary: This module provides a review of the binomial, geometric, hypergeometric, and Poisson probability distribution functions and their properties.

formula 1: Binomial

XX~B(n,p)B(n,p)

XX = the number of successes

nn = the number of independent trials

XX takes on the values x=x= 0,1, 2, 3, ...,nn

pp = the probability of a success

qq = the probability of a failure

p+q=1 q=1-pp+q=1 q=1-p

The mean is μ=npμ=np. The variance is σ2=npqσ2=npq.

formula 2: Geometric

XX~G (p)G(p)

XX = the number of trials until the first success (count the failures and the first success)

XX takes on the values xx= 1, 2, 3, ...

pp = the probability of a success

qq = the probability of a failure

p+q=1p+q=1

q=1-pq=1-p

The mean is μ= 1pμ= 1p

Τhe variance is σ2 = 1p (( 1p) -1) σ2 = 1p (( 1p) -1)

formula 3: Hypergeometric

XX~H (r, b, n)H(r,b,n)

XX = the number of items from the group of interest that are in the chosen sample.

XX may take on the values xx= 0, 1, ..., up to the size of the group of interest. (The minimum value for XX may be larger than 0 in some instances.)

rr = the size of the group of interest (first group)

bb= the size of the second group

nn= the size of the chosen sample.

n≤ r + b n r + b

The mean is: μ= n r r + b μ= n r r + b

The variance is: σ2= r b n ( r + b + n ) ( r + b ) 2 ( r + b - 1 ) σ2 r b n ( r + b + n ) ( r + b ) 2 ( r + b - 1 )

formula 4: Poisson

XX ~ P(μ)P(μ)

XX = the number of occurrences in the interval of interest

XX takes on the values xx = 0, 1, 2, 3, ...

The mean μμ is typically given. (λλ is often used as the mean instead of μμ.) When the Poisson is used to approximate the binomial, we use the binomial mean μ = npμ=np. nn is the binomial number of trials. pp = the probability of a success for each trial. This formula is valid when n is "large" and pp "small". nn "large" is about 100 and pp "small" may be something less than 0.1. If nn is large enough and pp is small enough then the Poisson approximates the binomial very well.

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