Mean of a Geometric Waiting Time

Condition on the outcome of the first trial to set up a recursive equation for the mean.

Solve the resulting equation $E[T]=1+(1-p)E[T]$.

1. 1. Condition on the first trial

   With probability $p$ the process stops after one trial; otherwise, one trial is used and the process restarts.

   $$E[T]=1+(1-p)E[T]$$

2. 2. Solve for the mean

   Rearranging gives $pE[T]=1$.

   $$E[T]=\frac{1}{p}$$

Answer: 1/p