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Joint probability distribution
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===Mixed case=== The "mixed joint density" may be defined where one or more random variables are continuous and the other random variables are discrete. With one variable of each type :<math> \begin{align} f_{X,Y}(x,y) = f_{X \mid Y}(x \mid y)\mathrm{P}(Y=y)= \mathrm{P}(Y=y \mid X=x) f_X(x). \end{align} </math> One example of a situation in which one may wish to find the cumulative distribution of one random variable which is continuous and another random variable which is discrete arises when one wishes to use a [[logistic regression]] in predicting the probability of a binary outcome Y conditional on the value of a continuously distributed outcome <math>X</math>. One ''must'' use the "mixed" joint density when finding the cumulative distribution of this binary outcome because the input variables <math>(X,Y)</math> were initially defined in such a way that one could not collectively assign it either a probability density function or a probability mass function. Formally, <math>f_{X,Y}(x,y)</math> is the probability density function of <math>(X,Y)</math> with respect to the [[product measure]] on the respective [[support (measure theory)|supports]] of <math>X</math> and <math>Y</math>. Either of these two decompositions can then be used to recover the joint cumulative distribution function: :<math> \begin{align} F_{X,Y}(x,y)&=\sum\limits_{t\le y}\int_{s=-\infty}^x f_{X,Y}(s,t)\;ds. \end{align} </math> The definition generalizes to a mixture of arbitrary numbers of discrete and continuous random variables.
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