I have $E[X|Y] = Y + \frac12$, and $E[Y] = \frac12$.
How can I use these two facts to demonstrate that $E[XY] = \frac34$?
I tried multiplying $E[X|Y]$ by $E[Y]$, but that didn't really get me anywhere...
2026-04-02 16:53:00.1775148780
How to use conditional expectation to find another expectation
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Using the tower rule and the fact that $E[YX\mid Y]=YE[X\mid Y]$:
$$\begin{align}E[XY] &= E[E[XY\mid Y]] \\[1ex] &= E[Y\,E[X\mid Y]] \\[1ex] &= E\left[Y\left(Y+\tfrac12\right)\right] \\[1ex] &= E\left[Y^2\right]+\tfrac12E[Y]\\[1ex] &= E\left[Y^2\right]+\tfrac14\end{align}$$
If $E\left[Y^2\right]=\tfrac12$ then, indeed, $E[XY]=\tfrac34.$