Question

Conditional variance

In the last example, we saw that the conditional distribution of X, which was a uniform over a smaller range (and in some sense, less uncertain), had a smaller variance, i.e., Var(X∣A)≤Var(X). Here is an example where this is not true. Let Y be uniform on {0,1,2} and let B be the event that Y belongs to {0,2}.

a) What is the variance of Y?

Var(Y)=

b) What is the conditional variance Var(Y∣B)?

Var(Y∣B)=

Answer 1

Therefore, V(Y|B) = E(Y² | B) + [ E(Y|B)]² = 2 - 1² = 2 - 1 = 1

V(Y|B) = 1

Clearly, from part a), and part b), we have

Var ( Y ) < Var(Y|B)