We have an unbalanced coin with 0.4 the probability of heads, and we toss it twice. Let $A$ be the random variable "number of heads on the first toss" and $B$ the random variable "number of total heads".
I was asked to find the correlation coefficient between $A$ and $B$ and then show numerically that $Cov(A,B)=VarA$.
Both were straightforward but now I am asked to show $Cov(A,B)=VarA$ without using numbers. How should I do this? With words or simply with general expressions of covariance and variance?
Let $A'$ be the number of heads on the second toss so that $A+A'=B$. Using bilinearity of covariance, and independence of $A$ and $A'$, we have
$$Cov(A,B) = Cov(A,A+A') = Cov(A,A)+Cov(A,A') = Var(A) + 0 = Var(A).$$