A fair coin is flipped 30 times. Let X denote the number of heads among the first 20 coin flips and Y denote the number of heads among the last 20 coin flips. Compute the correlation coefficient of X and Y.

Here's the set up given:

E[I_i]= P(ith coin is heads) = 1/2

Var[I_i]=E[I_i²]-(E[I_i])²

J₁=i=110 I_i

J₂=i=1120 I_i

J₃=i=2130 I_i

E[J₁]=i=110 E[I_i]

Var(J₁)=i=110 Var(I_i)

Similarly for J₂ and J_{3 to compute expectation and variance.}

Then X=J₁+J₂ and Y=J₂+J₃

Var(X)=Var(J₁+J₂)=Var(J₁)+Var(J₂)

Var(Y)=Var(J₂+J₃)=Var(J₂)+Var(J₃)

Cov(X,Y)=Cov(J₁+J₂, J₂+J₃)=Cov(J₁,J₂) + Cov(J₁,J₃)+Cov(J₂,J₂)+Cov(J₂,J₃)

Corr(X,Y)=Cov(X,Y)/(sqrt(var(X)sqrt(var(Y))

I know this going to be long computation. Can someone work this out using all the above notation? Thank you.