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Discrete Random Variables: Summary of the Discrete Probability Functions

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

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

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Formula 1: Binomial

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

XX = the number of successes in nn independent trials

nn = the number of independent trials

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

pp = the probability of a success for any trial

qq = the probability of a failure for any trial

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

The mean is μ=npμ=np. The standard deviation is σ=npqσ=npq.

Formula 2: Geometric

XX~G (p)G(p)

XX = the number of independent 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 for any trial

qq = the probability of a failure for any trial

p+q=1p+q=1

q=1-pq=1-p

The mean is μ= 1pμ= 1p

Τhe standard deviation is σ = 1p (( 1p) -1) σ = 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 standard deviation is: σ= r b n ( r + b + n ) ( r + b ) 2 ( r + b - 1 ) σ 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" (a general rule is that nn should be greater than or equal to 2020 and pp should be less than or equal to 0.050.05). If nn is large enough and pp is small enough then the Poisson approximates the binomial very well. The standard deviation is σ=μσ=μ.

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