Flipping a coin, length of runs

[bennyzadir]

Homework Statement

We flip a biased coin (the probability of a head is p, the probability of a tail is q=1-p). Denote X and Y the length of the first and the second run. A "run" is a maximal sequence of consecutive flips that are all the same. For example, if the sequence is HHHTHH... , then X=4, Y=1, if the sequence is THHTHTH..., then X=1, Y=2. Find the followings: E(X), E(Y), E(X^2), E(Y^2), Var(X), Var(Y), Cov(X, Y).

Homework Equations

The Attempt at a Solution

I am quite unsure, but I think we should use conditional expeted value conditioning with the result of the first toss. I suppose that to calculate Cov(X, Y) we should use the law of total covariance.

I would be really grateful if you could help me!

 

[mighty2000]

Hi there! I can help you with solving this problem.

To start, let's define some variables:
- X: the length of the first run
- Y: the length of the second run
- p: probability of getting a head
- q: probability of getting a tail (q=1-p)

Now, let's break down the problem and find the expected values and variances for each variable.

Expected value of X:
To find the expected value of X, we need to consider the possible outcomes of the first flip. If the first flip is a head, then the length of the first run is 1. If it is a tail, then the length of the first run is 0. Therefore, we can write the expected value of X as:

E(X) = 1 * p + 0 * q = p

Expected value of Y:
Similarly, we can find the expected value of Y by considering the possible outcomes of the second flip. If the first two flips are the same, then the length of the second run is 1. If they are different, then the length of the second run is 0. Therefore, we can write the expected value of Y as:

E(Y) = 1 * (p^2 + q^2) + 0 * (2pq) = p^2 + q^2

Expected value of X^2:
To find the expected value of X^2, we can use the formula E(X^2) = Var(X) + [E(X)]^2. We already know E(X), so we just need to find Var(X).

Variance of X:
To find the variance of X, we can use the formula Var(X) = E(X^2) - [E(X)]^2. Using the definition of X, we can write:

E(X^2) = 1^2 * p + 0^2 * q = p
Therefore, Var(X) = p - p^2 = p(1-p)

Expected value of Y^2:
Similarly, we can use the formula E(Y^2) = Var(Y) + [E(Y)]^2 to find the expected value of Y^2. We already know E(Y), so we just need to find Var(Y).

Variance of Y:
To find the variance of Y, we can use the formula Var(Y) = E(Y^