Question

Please explain the reasoning with calculation, thank you!

Question: If we toss a coin n times. x and y are number of heads and tails. What's the correlation coeffcient between x and y?

a)-1 b)0 c)1/2 d)1

Answer 1

Here,

X + Y = n

So,

E(X) + E(Y) = E(n) = n

On subtracting above two equations, we will have:

X - E(X) = - (Y - E(Y))

So,

Cov(X, Y) = E((X - E(X))(Y - E(Y)) = E(-(X - E(X))(X - E(X)) = - Var(X)

Also,

Var(X) = Var(Y)

Therefore,

Option A is correct.