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**1. 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]

**2. Homework Equations**

**3. 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]