# Expectation value of Geometric Distribution

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

### Proof

Since the probability of $X$ is given by the 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 expectation value is .
Example 2:
The graph below shows the geometric distribution for $p = 1/4$. The expectation value is .
Example 3:
The graph below shows the geometric distribution for $p = 1/8$. The expectation value is .
The smaller $p$ is, the more the graph becomes flat and the expected value increases.