Conditional expected value

• Aug 27th 2013, 02:37 AM
downthesun01
Conditional expected value
Can someone show me how $\displaystyle E[E[Y|X=x]]=E[Y]$

You can use your own joint distribution, or the uniform one I included below

Attachment 29067

$\displaystyle f(x,y)=\frac{2}{3}$

$\displaystyle f_{Y|X}(y|X=x)=\frac{1}{2-y}\text{ for } y\leq x\leq 2$
• Aug 27th 2013, 06:56 PM
chiro
Re: Conditional expected value
Hey downthesun01.

The general proof involves using the definitions (in continuous and discrete probability spaces) and basically "integrating out" the Y random variable after you integrate out the X random variable for a particular value.

Law of total expectation - Wikipedia, the free encyclopedia

For your particular case take the joint distribution, integrate out the x variable and then take the resulting function (which will depend on y) and integrate out the y variable.

Also note that when you integrate out you need to multiply the PDF by the variable (since E[X] = Integral x*f(x)dx for some RV x).