# Posterior probability

$f_{X\mid Y=y}(x)={f_X(x) L_{X\mid Y=y}(x) \over {\int_{-\infty}^\infty f_X(x) L_{X\mid Y=y}(x)\,dx}}$
• $f_X(x)$ is the prior density of X,
• $L_{X\mid Y=y}(x) = f_{Y\mid X=x}(y)$ is the likelihood function as a function of x,
• $\int_{-\infty}^\infty f_X(x) L_{X\mid Y=y}(x)\,dx$ is the normalizing constant, and
• $f_{X\mid Y=y}(x)$ is the posterior density of X.