Geometric Distribution - Related Distributions

Related Distributions

• The geometric distribution Y is a special case of the negative binomial distribution, with r = 1. More generally, if Y₁, ..., Y_r are independent geometrically distributed variables with parameter p, then the sum

follows a negative binomial distribution with parameters r and 1-p.

• If Y₁, ..., Y_r are independent geometrically distributed variables (with possibly different success parameters p_m), then their minimum

is also geometrically distributed, with parameter

• Suppose 0 < r < 1, and for k = 1, 2, 3, ... the random variable X_k has a Poisson distribution with expected value r k/k. Then

has a geometric distribution taking values in the set {0, 1, 2, ...}, with expected value r/(1 − r).

• The exponential distribution is the continuous analogue of the geometric distribution. If X is an exponentially distributed random variable with parameter λ, then

where is the floor (or greatest integer) function, is a geometrically distributed random variable with parameter p = 1 − e−λ (thus λ = −ln(1 − p)) and taking values in the set {0, 1, 2, ...}. This can be used to generate geometrically distributed pseudorandom numbers by first generating exponentially distributed pseudorandom numbers from a uniform pseudorandom number generator: then is geometrically distributed with parameter, if is uniformly distributed in .