Define rv Y with PY 0 1 Conditioning on Y is a way of cond

Define r.v. Y with P(Y = 0) = 1. Conditioning on Y is a way of conditioning on “no information” since P(Y = 0) = 1. Prove that for every x, P(X = x | Y) = P(X = x) = p(x)

Solution

The total probability of sample space is 1

Here the P(Y=0) = 1. There is no other value Y can take since P(0) = 1. Hence wee cannot condition on Y

P(X = x | Y) = P(X = x) = p(x)