Geometric and Negative Binomial Distributions

Geometric Distribution

X ~ Geo(p)

where p = probability of a success

Constraints of Geometric

• successive, independent trials

• all have the same probability

• need to know the number of trials until a success

Geometric Formulae

NEED TO KNOW

P(X = x)

• p(1-p)^x-1

P(X <= x)

• 1 - (1 - p)^x

P(X >= x)

• (1 - p)^x-1

Negative Binomial Distribution

X ~ NegB(r, p)

used to find the number of trials needed to achieve a fixed number of successes

Negative Binomial Formula

P(X = x) = (x - 1, r - 1)p^r(1 - p)^x-r

where (x - 1, r - 1) = (x - 1) C (r - 1)

Hypothesis Testing on Geometric Distribution

when the parameter p of a geometric distribution increases, the mean of the distribution decreases and vice versa

when setting up the hypothesis test to see if the parameter has increased you have to look to see if the test statistic falls in the lower end of the distribution