7  Bayes’ Theorem

7.1 Conditional continuous distributions

Definition 7.1 (Marginal probability density function) Given two continous random variables \(X\) and \(Y\) whose joint distribution is known, then the marginal probability density function is the integration of the joint probability distribution over \(Y\) and vice versa. That is \[ f_X(x)=\int_c^df(x,y)\dl3y,\quad f_Y(y)=\int_a^bf(x,y)\dl3x. \]

Definition 7.2 (Conditional Continous Distributions) Let \(X\) and \(Y\) be two random variables. The conditional probability density function of \(Y\) given the occurrence of the value \(x\) of \(X\) is written as \[ f_{Y\mid X}(y\mid x)=\frac{f_{X,Y}(x,y)}{f_X(x)} \] where \(f_{X,Y}(x,y)\) gives the joint density of \(X\) and \(Y\) while \(f_X(x)\) gives the marginal density of \(X\).