Penny flipped and uniform PDF generated

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SUMMARY

The discussion centers on calculating the mean and variance of a random variable X, which represents the sum of values generated from a uniform probability density function (PDF) over the interval [0,3] for each tail observed before the first head in a series of independent coin flips. The probability of flipping n tails before the first head is established as (0.5)^n for n ≥ 1, with the first flip having a probability of 1/2 for heads. The participants clarify that the expected value E[X] and variance Var(X) can be computed using the independence of the random variables involved, leading to the conclusion that the generation scheme must account for the possibility of zero tails. Two approaches to handle this case are discussed, with the second approach involving mixed discrete and continuous distributions.

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Homework Statement


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.

Homework Equations


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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alpines4 said:

Homework Statement


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.



Homework Equations





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.


You have one expression wrong: the probability of flipping n tails before the first head is 1/2 for n = 0 (i.e., the first toss is heads) and is (1/2)^n for n >= 1. Since it is possible that there are no tails before the first head, your generation scheme also needs to handle that case. I can see two "reasonable" ways: (i) ignore that case and do another sequence of flips; or (ii) when there are no tails, generate the single value "0".

In case (ii) the generated random variables would be mixed discrete and continuous, with a finite probability of the point 0. In case (i) the number of tails distribution would be P{Tails=n}= (1/2)^(n-1), n=1,2,... (the conditional probability of having first toss = tails).
So, case (i) is easier to work with and I will do that. You have [tex]EX =\sum_{n=1}^{\infty} P\{\text{Tails}=n\} E(X_1 + \cdots + X_n),[/tex]
and similarly for Var(X).

RGV
 
Thank you for your help, Ray.
 

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