Category:Definitions/Frequency Functions

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

Probability Mass Function

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

Let $X: \Omega \to \R$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.

Then the probability mass function of $X$ is the (real-valued) function $p_X: \R \to \closedint 0 1$ defined as:

$\forall x \in \R: \map {p_X} x = \begin{cases}

\map \Pr {\set {\omega \in \Omega: \map X \omega = x} } & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end{cases}$ where $\Omega_X$ is defined as $\Img X$, the image of $X$.

That is, $\map {p_X} x$ is the probability that the discrete random variable $X$ takes the value $x$.

Probability Density Function

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 $\lambda$.