Hello. I have the following problem to solve: prove that
Var(X)=E(Var(X¬Y)) + Var(E(X¬Y))
where ¬ denotes conditional probability. Sorry, I do not know how write on this LaTex thing
Hello,
Sorry I won't write it in Latex, I hope it will be enough.
The mean of a variable is not a variable anymore. It is a constant.
And so is the variance of a variable.
Now there are 2 properties of the mean and the variance. If a is a constant, then :
E{a}=a and Var{a}=0.
Hence E{Var(X|Y)}+Var{E(X|Y)}=Var(X|Y)+0=Var(X|Y)
So... is there some missing information or a mistake somewhere ? :s
True, but the conditional mean given a random variable is a random variable (there probably are another few chapters before you get to that cool notion ).
To prove the equality, one should make the definitions of the right-hand side explicit: since ,
Now, you should notice that two terms simplify, and if you remember that for any integrable , you'll notice that you're done.