The definition of the conditional p.d.f. of X given Y =
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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 seri
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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 ? , y0 + ]. Also, let y1 be such that f2(y1) > 0, and let A1 = [y1 ? , y1 + ]. Assume that f2 is continuous at both y0 and y1. Show that lim ?0 Pr(Y ? A0) Pr(Y ? A1) = 0. That is, the probability that Y is close to y0 is much smaller than the probability that Y is close to y1.
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