Stat. Geometric Distributions - Cooney

notes from textbook section 8.2: The Geometric Distribution


• a random variable X is geometric provided that the following conditions are met:
o each observation falls into one of just two categories, which for convenience we call “success” or “failure”
o the probability of a success, call it p, is the same for each observation
o the observations are all independent
o the variable of interest is the number of trials required to obtain the first success

• calculating geometric probabilities
o given that it’s geometric (see above), the probability that the first success occurs on the nth trial…
o P (X = n) = (1 – p)^(n-1) • p

• mean of a geometric random variable
o 1 / p
o a.k.a. the “expected value”

• variance
o (1-p) / p^2

• the probability that it takes more than n trials to see the first success is…
o P (X > n) = (1 – p)^n

• the sum of a geometric sequence
o (a) / (1 – r)
o a = probability (success)
o p = probability (failure)