The definition of the conditional p.d.f. of X given Y = y is arbitrary if f2(y) = 0. The reason that this causes no serious problem is that it is highly unlikely that we will observe Y close to a value y0 such that f2(y0) = 0. To be more precise, let f2(y0) = 0, and let A0 = [y0 ˆ’ ε, y0 + ε]. Also, let y1 be such that f2(y1) > 0, and let A1 = [y1 ˆ’ ε, y1 + ε]. Assume that f2 is continuous at both y0 and y1. Show that

That is, the probability that Y is close to y0 is much smaller than the probability that Y is close to y1.

Transcribed Image Text:

lim Pr(Ye Ao)-0.