<s>[INST] Finding conditional expectation on the basis of conditional probabilities Knowing that 


*

*$P(Y=1\mid X=5)=1/3$, 

*$P(Y=5\mid X=5)=2/3$. 


Calculate $E(Y\mid X=5)$ and $E(XY^2\mid X=5)$. How to solve this question? I have no idea whatsoever. 
 [/INST] Since the conditional probabilities sum up to $1$ then that is all for $Y\mid X=5$. So $$E[Y\mid X=5]=1\cdot P(Y=1\mid X=5)+5\cdot P(Y=5\mid X=5)=1\cdot\frac13+5\cdot\frac23=\frac{11}3$$ and 
\begin{align}E[XY^2\mid X=5]&=E[5Y^2\mid X=5]=5E[Y^2\mid X=5]\\[0.2cm]&=5\left((1)^2\cdot P(Y=1\mid X=5)+(5)^2\cdot P(Y=5\mid X=5)\right)\\[0.2cm]&=5\left(1\cdot\frac13+25\cdot\frac23\right)=\frac{255}3\end{align}
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