Conditional Expectation of pdf

Question

Wish to identify what I'm doing wrong when finding the $\operatorname E(X\mid Y=5)$ of the following: $$f(x, y)=\begin{cases} 1/6 & \text{if } 0<x<2, 0<y<6-3x \\ 0 & \text{otherwise} & \end{cases}.$$

My working: $$\begin{align}\operatorname E(X\mid Y=5)~&=~\int_0^2 x \frac{\frac{1}{6}}{f(y)} \, \mathsf d x\end{align}$$ where $$\begin{align}\mathsf f(y)~&=~\int_0^{\frac{-y+6}{3}} \frac{1}{6} \,\mathsf d x = \frac{6-y}{18}\end{align}$$

Therefore, $$\begin{align}\mathsf E(X\mid Y{=}5)~&=~\int_0^2 x \frac{\frac{1}{6}}{\frac{6-y}{18}}\,\mathsf d x \mathsf ~&=~\int_0^2 x \frac{\frac{1}{6}}{\frac{6-5}{18}}\,\mathsf d x~&=~\int_0^2 3x \,\mathsf d x \end{align} = 6$$

But then when I compute $\operatorname{Var}(X\mid Y=5),$ the answer is getting negative, which is impossible. So I think that I might have done something wrong when calculating the expectation.

My working for Variance: $$\operatorname{Var}(X\mid Y=5) = \operatorname E(X^2\mid Y=5) - (\operatorname E(X\mid Y=5) )^2 = 8 - 6^2 = -28$$

I appreciate any help.

Answer 1

Think about this geometrically: we've got $(X,Y)$ uniformly distributed over the right triangle with vertices at $(0,0)$, $(2,0)$, and $(0,6)$.

If we imagine sampling over and over from this region uniformly, we can picture $E(X\mid Y=5)$ as the average $x$ coordinate of the points that end up on the horizontal line $y=5$. All of these points will necessarily have $0 < x < 1/3$, since if $x \geq 1/3$ then $y = 5$ can't happen, and since the points are uniform over this line, this image suggests $E(X\mid Y=5) \stackrel {\text{?}}= 1/6$ (this is the midpoint of the range of $x$ points that can end up on this line).

Checking this with calculus, the issue with what you did is the limits of integration for $x$ also depend on $y$. If $y = 5$ is observed then the biggest that $x$ can be is $1/3$, so really $$ E(X\mid Y=5) = \int_0^{1/3}x \frac{3}{6-y}\,\text dx = 3 \cdot \frac 12 x^2\bigg|_{x=0}^{x=1/3} = \frac 16. $$

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Another quick sanity check is that $E(X\mid Y)$ has to be within the range of $X$, so $E(X\mid Y=5)\neq 6$ can be recognized as incorrect before the negative variance shows up.