covariance of 2 poisson RVs

Question

Im trying to calculate the covariance of x & y.

Heres what I am given: z1 and z2 are independent poisson random variables with parameters 3 for Z1 and 5 for Z2.

x= (z1+z2) y=(z1-z2)

I have that x and y are Poi(7) and Poi(-2) and are dependent.

I feel like I am overlooking something very simple(or making a stupid mistake), but cant figure it out. Any hints/help would be appreciated!

Answer 1

I will use caps for the random variables. The covariance of $X$ and $Y$ is $$E(XY)-E(X)E(Y).$$ The expectations of $X$ and $Y$ are easy to compute using the linearity of expectation. As to $E(XY)$, this is $E(Z_1^2)-E(Z_2^2)$. You can compute the $E(Z_i^2)$ since you probably know the variances of the $Z_i$. Then use $E(Z_i^2)=\text{Var}(Z_i)+(E(Z_i))^2$.

Answer 2

There is no such thing as a Poisson distribution with parameter $-2$. $z_1 - z_2$ is not Poisson.

As for the covariance, just expand: the covariance of $X$ and $Y$ is linear in each of $X$ and $Y$. So $$\text{Cov}(z_1 + z_2, z_1 - z_2) = \text{Cov}(z_1, z_1) + \text{Cov}(z_2, z_1) - \text{Cov}(z_1, z_2) - \text{Cov}(z_2, z_2) = \ldots$$