Flipping a biased coin and finding the covariance and correlation

Question

You flip a biased coin 15 times, independently. The coin has probability p of being a heads. Let X be the number of heads in the first 10 coin flips and Y be the total number of heads in the last 10 coin flips.

Compute Cov(X, Y ) and their correlation ρ.

1. i got 5p(1-p) for the cov but im not sure if thats right and not sure how to find the correlation of p

Answer 1

You are right about 5p(1-p). Let the sum of the first 5 flips be a, the next five b, and the next five c

X = a+b

Y=b+c

Then, cov(X,Y) = cov(a+b,b+c) = cov(a,b) + cov(a,c) + cov(b,b) + cob(b,c) = 0 + 0 + 5p(1-p) + 0 = 5p(1-p)

Correlation = cov(X,Y) /sqrt(var(X)var(Y)) = 5p(1-p)/(10p(1-p)) = 1/2