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Homework Statement
Let [itex]X[/itex] and [itex]Y[/itex] be independent Bernoulli RV's with parameter [itex]p[/itex]. Find,
[tex]\mathbb{E}[X\vert 1_{\{X+Y=0\}}][/tex] and [tex]\mathbb{E}[Y\vert 1_{\{X+Y=0\}}][/tex]
Homework Equations
The Attempt at a Solution
I'm trying to show that,
[tex]\mathbb{E}[X+Y\vert 1_{\{X+Y=0\}}] = 0[/tex]
by,
[tex]
\begin{align*}
\mathbb{E}[X+Y\vert 1_{\{X+Y=0\}}] &= \frac{\mathbb{E}[(X+Y)1_{\{X+Y=0\}}]}{\mathbb{P}[X+Y=0]} \\
&= \frac{0}{(1-p)^2} \\
&= 0
\end{align*}
[/tex]