Marlene Brooks

## Answered question

2022-10-12

Recursive proof of expectation of geometric distribution?
If $T\sim \mathsf{G}\mathsf{e}\mathsf{o}\left(p\right)$ ($0) then $\mathbf{E}T=1/p$ is well known.
Suppose that I toss a sequence of coins that come up heads with probability p. On average, how many coins do I have to toss before I see a heads?

### Answer & Explanation

Jovanni Salinas

Beginner2022-10-13Added 18 answers

Step 1
We can condition on the first toss, let H denote the event that the first toss is a head.
$\begin{array}{rl}E\left[T\right]& =E\left[T|H\right]E\left[H\right]+E\left[T|{H}^{c}\right]E\left[{H}^{c}\right]\\ & =p+\left(1+E\left[T\right]\right)\left(1-p\right)\\ & =1+\left(1-p\right)E\left[T\right]\end{array}$
Step 2
Solving for E[T] gives us $\frac{1}{p}$.
Notice that $E\left[T|{H}^{c}\right]=1+E\left[T\right]$ as the first toss and the remaining tosses are independent.

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