(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A penny is flipped until we see the first head, and flips are assumed to be independent. For each tail we observe before the first head, the value of a continuous random variable with uniform PDF over the interval [0,3] is generated. Let the RV X be the sum of all the values obtained before the first head. We want to find the mean and variance of X.

2. Relevant equations

3. The attempt at a solution

Assuming n tails before the first head, we have E[X] = E[X_1 + ... + X_n] = E[X_1] + ... + E[X_n], and Var(X) = Var(X_1 + ... + X_n) = Var(X_1) + ... + Var(X_n) as the X_i are independent so there are no covariance terms.

Since each X_i has a discrete uniform distribution, the pdf of each X_i is 1/4 for X_i = 0,1,2, or 3.

Also, the probability of flipping n tails before the first head is (0.5)^n.

However, I'm not sure how to put all of this together. I think the mean of X will be (1/4) + ...+ (1/4) n times, but this doesn't take into account the probability of flipping n tails. Any help would be appreciated.

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# Homework Help: Penny flipped and uniform PDF generated

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