Expectation value of Geometric Distribution

Last updated: Jan. 3, 2019
  Let $X$ be a random variable with the geometric distribution
Geometric distribution
, where $k=1,2,\cdots. $ The expectation value of $X$ is
Expectation value of geometric distribution
, where $0 \lt p \lt 1$.

  Proof

Since the probability of $X$ is given by the geometric distribution,
Geometric distribution
, the expectation value is
.
  To find the sum in the right-hand side, we use Taylor expansion of function $1/(1-x)$,
. The derivative of the left-hand side is
, and that of the right-hand side is
. We thus have
. By variable transformation, $x= 1-p$, this equation becomes
.
  Substituting this into $(1)$, we obtain
.
Examples
Example 1:
  The graph below shows the geometric distribution for $ p = 1/2 $.
The graph of the geometric distribution of p=1/2
The expectation value is
.
Example 2:
  The graph below shows the geometric distribution for $ p = 1/4 $.
The graph of the geometric distribution of p=1/4
The expectation value is
.
Example 3:
  The graph below shows the geometric distribution for $ p = 1/8 $.
The graph of the geometric distribution of p=1/8
The expectation value is
.
  The smaller $p$ is, the more the graph becomes flat and the expected value increases.