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Math Help - Finding the expected value of r^X

  1. #1
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    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?
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    MHF Contributor chisigma's Avatar
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    Re: Finding the expected value of r^X

    Quote Originally Posted by mklee90 View Post
    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...

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

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

    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

    \chi \sigma
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  3. #3
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    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 :

    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.
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