L09.2 Conditioning A Continuous Random Variable on an Event
In this segment, we pursue two themes. Every concept has a conditional counterpart. We know about PDFs, but if we live in a conditional universe, then we deal with conditional probabilities. And we need to use conditional PDFs. The second theme is that discrete formulas have continuous counterparts in which summations get replaced by integrals, and PMFs by PDFs. So let us recall the definition of a conditional PMF, which is just the same as an ordinary PMF but applied to a conditional universe. In the same spirit, we can start with a PDF, which we can interpret, for example, in terms of probabilities of small intervals. If we move to a conditional model in which event A is known to have occurred, probabilities of small intervals will then be determined by a conditional PDF, which we denote in this manner. Of course, we need to assume throughout that the probability of the conditioning event is positive so that conditional probabilities are well-defined. Let us now push th...