# Thread: Finding the expected value of r^X

1. ## Finding the expected value of r^X

Hello, I have recently enrolled in a 3rd year Introductory Probability course.
I'm having a pretty tough time with the course so I came to this forum again to seek some guidance.

When looking at geometric random variables I was asked to find the expected value of r^X, where X is a geometric rv.

I know that E(X) = Sum(n*(1-p)^(n-1)*p) in a geometric series... but it confuses me when I have to find E(r^X).

What do I have to do in this case?

2. ## Re: Finding the expected value of r^X

Originally Posted by mklee90
Hello, I have recently enrolled in a 3rd year Introductory Probability course.
I'm having a pretty tough time with the course so I came to this forum again to seek some guidance.

When looking at geometric random variables I was asked to find the expected value of r^X, where X is a geometric rv.

I know that E(X) = Sum(n*(1-p)^(n-1)*p) in a geometric series... but it confuses me when I have to find E(r^X).

What do I have to do in this case?
If X is a 'geometric random variable', the its probability function is...

$\displaystyle P\{X=k\}= p\ (1-p)^{k-1}\ ,\ k \ge 1$ (1)

... so that 'by definition' is, under the hypothesis that $\displaystyle |r\ (1-p)|<1$, ...

$\displaystyle E\{r^{X}\} = \frac{p}{1-p}\ \sum_{k=1}^{\infty} \{r\ (1-p)\}^{k} = \frac{r\ p}{1-r\ (1-p)}$ (2)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. ## Re: Finding the expected value of r^X

It's not a matter of definition related to a geometric distribution. It's just that for any 'correct' function f and, for example, a discrete random variable X, with probability function p, we have :

$\displaystyle E[f(X)]=\sum_{k=0}^\infty f(x)p(x)$

If X takes integer and positive values. Otherwise, you'll have to change the range of k, but that's all.