Finding covariance given the following data

Question

We know that:

$P(Z>0|X=0)=0$

We also know that:

$P(X>0)=0.5$

$P(Z>0|X>0)=0.5$

$E(X|X>0 \wedge Z=0)=2$

$E(X|X>0 \wedge Z>0)=4$

$E(Z|Z>0)=4$

$cov(X,Z|X>0 \wedge Z>0)=c$.

Find $cov(X,Z)$ in terms of parameter c. I am overwhelmed by the amount of data in this question, how to solve it?

Answer 1

Well, you seek to find $$\begin{align}\mathsf {Cov}(X,Z) =&~ \mathsf E(XZ)-\mathsf E(X)\mathsf E(Z)\end{align}$$

So you will need to find those three terms from the conditional expectations and probabilities you've been given.   The Law of Total Expectation will come into play.

$$\begin{align}\mathsf E(X) =&~ \mathsf E(X\mid X>0)\mathsf P(X>0)+\mathsf E(X\mid X=0)\mathsf P(X=0)\\[1ex] = & ~ 0.5~\mathsf E(X\mid X>0) \\[1ex] =&~ 0.5~\big(\mathsf E(X\mid X>0\cap Z=0)\mathsf P(Z=0\mid X>0)+\mathsf E(X\mid X>0\cap Z>0)\mathsf P(Z>0\mid X>0)\big)\end{align}$$

And so forth...