Category:Definitions/Probability Density Functions

This category contains definitions related to Probability Density Functions.
Related results can be found in Category:Probability Density Functions.

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be an absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $P_X$ be the probability distribution of $X$.

Let $\map \BB \R$ be the Borel $\sigma$-algebra of $\R$.

Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.

We define the probability density function $f_X$ by:

$\ds f_X = \frac {\d P_X} {\d \lambda}$

where $\dfrac {\d P_X} {\d \lambda}$ denotes the Radon-Nikodym derivative of $P_X$ with respect to $\lambda$.