Penny flipped and uniform PDF generated

In summary, you can find the mean and variance of a sum of discrete uniform random variables by taking the sum of their individual pdfs. For each tail, the probability of flipping n heads is (0.5)^n, and the mean is (1/4) + ...+ (1/4) n times.
  • #1
alpines4
4
0

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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  • #2
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
 
  • #3
Thank you for your help, Ray.
 

1. What is a "Penny flipped and uniform PDF generated"?

A "Penny flipped and uniform PDF generated" refers to a random experiment in which a penny is flipped and the outcome is recorded as either heads or tails. This experiment can be used to generate a uniform probability distribution function (PDF) where each outcome has an equal chance of occurring.

2. How is a uniform PDF generated from flipping a penny?

A uniform PDF is generated from flipping a penny by recording the outcomes of multiple flips and plotting the frequency of each outcome. As the number of flips increases, the frequency of each outcome should approach a uniform distribution.

3. What is the significance of using a penny to generate a uniform PDF?

Using a penny to generate a uniform PDF is significant because it demonstrates the concept of a random experiment and its relationship to a uniform distribution. The simplicity of flipping a penny also makes it easy to understand and replicate, making it a useful tool in teaching probability and statistics.

4. Can a different coin be used to generate a uniform PDF?

Yes, any coin with two distinct sides can be used to generate a uniform PDF. However, the size and weight of the coin may affect the results, so it is important to use the same coin for consistency.

5. How is a uniform PDF useful in scientific research?

A uniform PDF is useful in scientific research as it can be used to model and analyze random events and phenomena. It is also a fundamental concept in statistics and probability, which are essential tools in many scientific fields such as biology, physics, and psychology.

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