Suppose I have three real-valued random variables A, B and S with poisson or negative binomial distribution. Let (⋅) denote the covariance operator. How do I evaluate Cov(B, AS)?
I have
Cov(B, AS) = E[B * AS] - E[B]E[AS]
E[AS] = Cov(A, S) + E[A]E[S]
Cov(B, AS) = E[B * A * S] - E[B](Cov(A,S) + E[A]E[S]) = E[B * A * S] - E[B]*Cov(A,S) - E[B]E[A]E[S]
I have known that Cov(A,S) > 0, and E[A], E[B] andE[S] are all greater than zero.
With this, I cannot decide the sign of Cov(B,AS). Is this right?