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March 13th, 2016, 06:38 PM   #1
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Uniform Joint Distribution

Suppose Y is uniformly distributed on (0, 1), and that the conditional distribution of X given that Y = y is uniform on (0, y). Find E[X] and Var(X).

Not sure how to put this one together since it is a joint distribution. Here is my attempt so far.

$\displaystyle f_Y(y)=1$ , for $\displaystyle 0 \leq y \leq 1$
for $\displaystyle X$ to be uniform on $\displaystyle (0,y)$

$\displaystyle f_{x|y}(x,y)=1/y $ , for $\displaystyle 0 \leq x \leq y$

so does $\displaystyle E[X] = \frac{y-0}{2} = \frac{y}{2}=\frac{1}{2}$ ??

Am I starting this off right?

Thank you for any help!
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March 14th, 2016, 03:57 PM   #2
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$\displaystyle E(X|y)=\frac{1}{y}\int_0^y xdx=\frac{y}{2}$
$\displaystyle E(X)=\int_0^1 E(X|y)dy=\frac{1}{4}$
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