Problem
- Let $X_1,X_2,…$ be independent random variables with $E(X_i)=a, Var(X_i)=b$ for $i\geq1$,
- and $N\geq0$ an integer-valued random variable with $E(N) = c, Var(N) = d^2$ independent of the sequence $\{X_i\}$.
- Let $Y_N=X_1+···+ X_N$ with $Y_0 = 0$.
- Compute $Cov(Y_N, N)$.
My Approach
I know that
$$Cov(Y_N, N) = E[Y_NN] – E[Y_N]E[N]$$
Using the Law of Total Expectation I have determined the $E[Y_N] = ac$ and the expected value of $N$, $E[N]=c$ but I don't know how to find the $E[Y_NN]$.
Best Answer
Observe that $$E(NY_N \, | \, N = n) = E \left( n \sum_{k=1}^n Y_k \right) = n^2 a$$ So that $(E(NY_N \, | \, N) = N^2a$. By the law of total expectation: $$E(NY_N) = E(E(NY_N \, | \, N)) = E(N^2 a) = a (Var(N) + E(N)^2) = a(c^2 + d^2)$$ Using $E(Y_N) = ac$ which you computed previously, we get: $$Cov(Y_N, N) = E(Y_N N) - E(Y_N)E(N) = a(c^2 +d^2) - ac^2 = ad^2$$