Category:Frequency Functions
This category contains results about Frequency Functions.
Definitions specific to this category can be found in Definitions/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 $\lambda$.
Subcategories
This category has only the following subcategory.