Question

3. From the set 1,2,.. 10 we randomly draw, without replacement, two numbers. Let X be the smaller and Y the larger of the two values. Calculate E(XY) and E(XY X|X).

Answer 1

$P(X=x|Y=y)=\frac{1}{y},~x=1,2,...,y~\\ (Since~X~is~smaller~than~Y~so~if~Y=y~then~X~may~take~any~one~of~1,2,...,y)\\ =0~otherwise\\ E(X|y)=\frac{1}{y}\sum_{x=1}^yx=\frac{y+1}{2}\\ E(X|Y)=\frac{Y+1}{2}\\Similarly~P(Y=y|X=x)=\frac{1}{10-x},~y=x+1,x+2,...,10\\ =0~otherwise\\$

$E(Y|x)=\frac{1}{10-x}\sum_{y=x+1}^{10}y=\frac{1}{10-x}\left(\frac{10*11}{2}-\frac{x(x+1)}{2} \right )\\ =\frac{1}{2(10-x)}\left((10-x)(11+x) \right )=\frac{11+x}{2}\\ E(XY+X|X)=XE(Y|X)+X=X\times \frac{11+X}{2}+X=\frac{X(X+13)}{2}$