Question

Flip a fair coin 6 times. Let X be the number of heads in the rst 4 ips, and Y be the number of heads on the last 4 ips. Find Cov(X; Y ).

Answer 1

Cov ( X, Y ) = 1/2

Explanation:

X = Number of heads in the first 4 flips

Y = Number of heads in the last 4 flips

Given that X and Y are binomial variables hence

P( probability ) = 1/2

Find Cov( X; Y )

xi = result of the ith flip ∴ X = x1 + x2 + x3 + x4

yj = result of the jth flip ∴ Y = y3 + y4 + y5 + y6

covariance of xi and yi = 1/2 * 1/2 = 1/4 when i = j and it is = 0 when i ≠ j

hence Cov( X; Y ) can be expressed as

Cov( X; Y ) = ∑
`_(i)`^4 ∑
`_(j)`^6 ∴ Cov( Xi , Yj ) = 2/4 = 1/2 ( given that i = j )