User:Caliburn/sandbox/findsourceforthese

These are in note form for now.

Definition (Almost Surely Constant Random Variable)
Let $X$ be a random variable.

Let:


 * $\map \Pr {X = c} = 1$

for some $c \in \R$.

We say that $X$ is almost surely constant.

Note that this is distinct from $X$ always taking the value $c$. Rather it takes the value $c$ with probability $1$.

For example if $X \sim \ContinuousUniform 0 1$ then $\map { {\mathbf 1}_{\Q} } X$ takes the value $0$ with probability $1$ and so is almost surely constant, yet it is possible for it to take the value $1$. It just has probability $0$ of doing so.

$X$ might be simply, perhaps misleadingly, called constant.

Definition (Negative Random Variable)
Let $X$ be a random variable.

Let:


 * $\map \Pr {X < 0} = 1$

We say that $X$ is negative.

$X$ might take non-negative values but with probability $0$.

Definition (Non-Positive Random Variable)
Let $X$ be a random variable.

Let:


 * $\map \Pr {X \le 0} = 1$

We say that $X$ is non-positive.

$X$ might take positive values but with probability $0$.

Definition (Positive Random Variable)
Let $X$ be a random variable.

Let:


 * $\map \Pr {X > 0} = 1$

We say that $X$ is positive.

$X$ might take non-positive values but with probability $0$.

Definition (Non-Negative Random Variable)
Let $X$ be a random variable.

Let:


 * $\map \Pr {X \ge 0} = 1$

We say that $X$ is non-negative.

$X$ might take negative values but with probability $0$.

Definition (Support (Discrete))
Let $X$ be a discrete random variable.

We define the support of $X$, $\map {\operatorname {supp} } X$, by:


 * $\map {\operatorname {supp} } X = \set {x \in \Img X \mid \map \Pr {X = x} \ne 0}$

Definition (Support (Continuous))
Let $X$ be a continuous random variable.

Let $f_X$ be the probability density function of $X$.

We define the support of $X$, $\map {\operatorname {supp} } X$, by:


 * $\map {\operatorname {supp} } X = \set {x \in \Img X \mid \map {f_X} x \ne 0}$