Conditional expected value meaning problem

Question

I have two random variables X,Y with the same following distributions :

$$P(X=0)=P(Y=0)=1-p, P(X=1)=P(Y=1)=p$$

I want to calculate $E(X|1_{\{X+Y=0\}})$.

So Let's define random variable $Z=1_{\{X+Y=0\}}$. Then

$P(Z=1)=(1-p)^2$ and $P(Z=0)=1-(1-p)^2$

$E(X|1_{\{X+Y=0\}})=E(X|Z)=\frac{1}{P(Z)} \cdot \int_{Z}X \;dP$.

So now i have a little trouble. What exactly is $P(Z)$ ? I think that i cant even calculate that. This problem is like : Let $X \sim U[0,1]$ find $P(X)$ which is making no sense.

Answer 1

You have not stated enough information to solve the problem. In particular, you have only specified the marginal distributions of $X$ and $Y$, but not their joint distribution. Two cases satisfying your description are when $X$ and $Y$ are independent, and when $X=Y$ with probability $1$. They give rise to different answers.