Discuss distribution function of mixed random variable? - Answers

In some situationsX is continuous but Y is discrete. For example, in a logistic regression, one may wish to predict the probability of a binary outcome Y conditional on the value of a continuously-distributed X. In this case, (X, Y) has neither a probability density function nor a probability mass function in the sense of the terms given above. On the other hand, a "mixed joint density" can be defined in either of two ways:Formally, fX,Y(x, y) is the probability density function of (X, Y) with respect to the product measure on the respective supports of X and Y. Either of these two decompositions can then be used to recover the joint cumulative distribution function:The definition generalizes to a mixture of arbitrary numbers of discrete and continuous random variables.