Two Expected value definitions of the geometric random variable

Question

Ok so I'm looking at my book and it defines the geometric distribution to be $\sum_{n=1}^{\infty}p(1-p)^{n-1}$. My book says the expected value of a geometric random variable is $\dfrac{p}{q}$. It proves it using the probablity generating function for the geometric random variable. However, I have seen another expected value of a geometric random variable to be $\dfrac{1}{p}$. How are these two definitions related and when do I use one or the other?

Answer 1

The geometric random variable is either

• The number of trials needed to get one success, in $\{1,2,3,\ldots\}$; or
• The number of failures before the first success, in $\{0,1,2,3,\ldots\}$.

These are two different things. You have to look at which of the two the book in question has in mind. The first one has expeccted value $1/p$, where $p$ is the probability of success on each trial. The second has expected value $(1/p)-1$ $=(1-p)/p$.

The second one has an infinitely divisible distribution; the first is not.