Bernoulli

• have a single event with two outcomes: "success" and "failure"
• probability p for "success" (1)
• q = 1−p probability for "failure" (0)

X P(X) 0 1−p 1 p

      μ = p,      &sigma² = p(1−p)

Binomial

• n independent events with two outcomes
• constant p probability for "success"
• constant q = 1−p for "failure"

      X can be any of 0, 1, 2, ..., n

      μ = np,      &sigma² = np(1−p)

            Binomial Calculator

Geometric

• n independent events with two outcomes
• constant p probability for "success"
• constant q = 1−p for "failure"
• want first time a "success" occurs—think Russian Roulette

      X can be any of 1, 2, 3, ....

      μ = 1/p,      &sigma² = (1−p)/p²

            Geometric Calculator

Negative Binomial

• n independent events with two outcomes
• constant p probability for "success"
• constant q = 1−p for "failure"
• want the trial that the r^th "success" occurs (note if r=1, then becomes Geometric)

      X can be any of r, r+1, r+2, ....

      μ = r/p,      &sigma² = r(1−p)/p²

Hypergeometric

• events are not independent
• have a population with N total items consisting of two types
• of the N items, k are the designated type
• have r = N−k is the number of remaining items
• extract a sample of n items without replacement
• have x is the number of the designated type in the sample
• have y = n−x is the number in the sample that is not of the designated type

P(X=x)  =  kCx • rCy   =  kCx • N−kCn−x NCn NCn       X can be any of 0, 1, 2, ... , smaller of k or n       μ = n k ,      &sigma2 = n k • N−k • N−n N N N N−1

⌈ N ⌉ k   r = N−k |   | x   y = n−x ⌊ n ⌋

            Hypergeometric Calculator

Poisson

• n independent events with two outcomes
• constant p probability for "success"
• constant q = 1−p for "failure"
• like the Binomial Distribution except large number n and extremely small probability p but with the product np = λ being reasonable

P(X=x)  =	e−λλx
	x!

      X can be any of 0, 1, 2, ....

      μ = &lambda,      &sigma² = λ

            Poisson Calculator