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I am currently taking a probability course and I am stuck on a supposedly easy discrete probability question here:

Problem: Consider the experiment of rolling a fair die independently until the same number/face occurs 2 successive times and let $X$ be the trial on which the repeat occurs, e.g. if the rolls are $2,3,4,5,1,2,4,5,5$, then $X=9$.

a. find the probability function $f(x) = P(X=x)$

b. compute EX

Attempt at a solution: I know $X$ is discrete probability distribution, and that we are dealing with independent events. However, $X$ can be anything, up to infinity, or it may never happen where there are two successive values. Here's what I got:

obviously, the answer is a geometric distribution as the answer is:

$P(X=x) = f(x) = (5/6)^(x-1) * (1/6)$ for $x = 0,1,2,3,...$ and $0$ otherwise but I'm stuck here. please help

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    X=9, it's the 9th trial.. Sorry about that typo2012-04-30
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    Also what exactly are you stuck on? You have derived that it's geometric. So you need help with $E(X)$?2012-04-30
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    Yea, the EX for this function and if this is indeed a geometric distribution, I have no way of telling2012-04-30
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    It's the "obviously, the answer is..." part that confounded me2012-04-30
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    I'm seconding guessing my "obvious" insight now...2012-04-30
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    I think I got it, thanks2012-04-30
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    Your $f(x)$ is *almost* right. The first roll is free, and $f(0)=0$. Then $f(2)=1/6$, $f(3)=(5/6)(1/6)$, $f(4)=(5/6)^2(1/6)$, and in general $f(x)=(5/6)^{x-2}(1/6)$ for $x\ge 2$.2012-04-30
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    Right, I'm seconding guess now because f(x) does not equal 1. Sorry for the confusion2012-04-30

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