L10.9 Mixed Bayes Rule
We have seen two versions of the Bayes rule-- one involving two discrete random variables, and another that involves two continuous random variables. But there are many situations in real life when one has to deal simultaneously with discrete and continuous random variables. For example, you may want to recover a discrete digital signal that was sent to you, but the signal has been corrupted by continuous noise so that your observation is a continuous random variable. So suppose that we have a discrete random variable K, and another continuous random variable, Y. In order to get a variant of the Bayes rule that applies to this situation, we will proceed as in the more standard cases. We will use the multiplication rule twice to get two alternative expressions for the probability of two events happening. We will equate those expressions, and from these, derive a version of the Bayes rule. So we will look at the probability that the discrete random variable takes on a certa...