Discrete random variable

Geometric distribution

The geometric distribution describes the number of failures of independent Bernoulli trials with success probability before the first success. #m/def/prob It has the probability mass function

for .

Proof

Let be independent for . Then

as claimed.

This is related to the Negative binomial distribution, which is the sum of i.i.d. geometric variables.

Properties

Let and

  1. Expectation:
  2. Variance:
  3. Moment-generating function:
Proof of 1

Invoking the expansion for a geometric series

as claimed, proving ^P1.

Alternately we may invoke conditional expected value. Let denote the event that the first trial is successful. Then noting that , we have

whence

proving ^P1.

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