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

Binomial

X ~ B ( n , p )

X = the number of successes in n independent trials

n = the number of independent trials

X takes on the values x = 0,1, 2, 3, ..., n

p = the probability of a success for any trial

q = the probability of a failure for any trial

p + q = 1 q = 1 - p

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

Formula

Geometric

X ~ G ( p )

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

X takes on the values x = 1, 2, 3, ...

p = the probability of a success for any trial

q = the probability of a failure for any trial

p + q = 1

q = 1 - p

The mean is μ = 1 p

Τhe standard deviation is σ = 1 p ( ( 1 p ) - 1 )

Formula

Hypergeometric

X ~ H ( r , b , n )

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

X may take on the values x = 0, 1, ..., up to the size of the group of interest. (The minimum valuefor X may be larger than 0 in some instances.)

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

b = the size of the second group

n = the size of the chosen sample.

n r + b

The mean is: μ = n r r + b

The standard deviation is: σ r b n ( r + b - n ) ( r + b ) 2 ( r + b - 1 )

Formula

Poisson

X ~ P(μ)

X = the number of occurrences in the interval of interest

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

The mean μ is typically given. ( λ is often used as the mean instead of μ .) When the Poisson is used to approximate the binomial, we use the binomialmean μ = n p . n is the binomial number of trials. p = the probability of a success for each trial. This formula is valid when n is "large" and p "small" (a general rule is that n should be greater than or equal to 20 and p should be less than or equal to 0.05 ). If n is large enough and p is small enough then the Poisson approximates the binomial very well. The variance is σ 2 = μ and the standard deviation is σ = μ

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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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