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March 13th, 2016, 06:38 PM  #1 
Senior Member Joined: Jan 2014 Posts: 196 Thanks: 3  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_{xy}(x,y)=1/y $ , for $\displaystyle 0 \leq x \leq y$ so does $\displaystyle E[X] = \frac{y0}{2} = \frac{y}{2}=\frac{1}{2}$ ?? Am I starting this off right? Thank you for any help! 
March 14th, 2016, 03:57 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,856 Thanks: 745 
$\displaystyle E(Xy)=\frac{1}{y}\int_0^y xdx=\frac{y}{2}$ $\displaystyle E(X)=\int_0^1 E(Xy)dy=\frac{1}{4}$ 

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distribution, joint, uniform 
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