Given a probability space \(\left(\Omega,\mathcal{F},\mathbb{P}\right)\) and a measurable space \(\left(S,\mathcal{S}\right),\) a random variable is a function \(X:\Omega\rightarrow S\) such that:

\[\left\{\omega\in\Omega:X\left(\omega\right)\in A\right\}\in\mathcal{F}\,\,\forall\,A\in\mathcal{S}.\]

\(S\) is called the state space of \(X\) and \(\mathcal{S}\) is its corresponding \(\sigma\)-algebra.

A second probability measure is often constructed when referring to the random variable \(X\) and the values in the state space \(S\) so that the underlying probability space \(\left(\Omega,\mathcal{F},\mathbb{P}\right)\) and the original probability measure \(\mathbb{P}\) do not need to be continually referenced.

Given a random variable \(X\) defined with respect to an underlying probability space \(\left(\Omega,\mathcal{F},\mathbb{P}\right)\) and measurable space \(\left(S,\mathcal{S}\right),\) the pushforward probability measure is a probability measure \(\mathbb{P}_X\) on \(\left(S,\mathcal{S}\right)\) such that:

\[\mathbb{P}_X\left(X\in A\right) = \mathbb{P}\left(\left\{\omega\in\Omega:X\left(\omega\right)\in A\right\}\right)\,\,\forall\,A\in\mathcal{S}.\]

The pushforward probability measure \(\mathbb{P}_X\) assigns probabilities to measurable subsets of the state space, \(A\in\mathcal{S},\) and defines the probability distribution of \(X\) over the state space \(S.\) The complete probability space is \(\left(S,\mathcal{S},\mathbb{P}_X\right).\)

The support of the probability distribution \(\mathbb{P}_X\) is the subset of the state space \(\mathcal{X}\subseteq S\) such that the following properties are satisfied:

\[\mathbb{P}_X\left(X\in A\right)>0\,\,\forall\,A\subseteq\mathcal{X},A\neq\varnothing\] \[\mathbb{P}_X\left(\mathcal{X}\right)=1\]

Given a random variable \(X\) defined on the measurable space \(\left(S,\mathcal{S}\right),\) \(X\) is a discrete random variable if there is an injective function between the state space \(S\) and the set (or subset) of natural numbers \(\mathbb{N}.\) That is, \(S\) is countable:

\[S=\{x_1,\ldots,x_n\}\,\mathrm{(finite), or}\] \[S=\{x_1,\ldots\}\,\mathrm{(infinite)}\]

In this scenario \(S\) is referred to as a discrete state space.

Let \(X\) be a discrete random variable defined on the probability space \(\left(S,\mathcal{S},\mathbb{P}_X\right).\) The probability mass function (pmf) is a function \(p_X:S\rightarrow\left[0,1\right]\) defined by:

\[p_X\left(x\right)=\mathbb{P}_X\left(X=x\right)\,\,\forall\,x\in S\]

The pmf \(p_X\left(x\right)\) assigns probability to each element \(x\in S.\) Generally, the pmf is defined to be zero for all values \(x\in\mathbb{R}\setminus S\) so that \(p_X\left(x\right)\) is defined for all \(x\in\mathbb{R}.\)

The pmf \(p_X\left(x\right)\) has the following properties:

\[p_X\left(x\right)\geq 0\,\,\forall\,x\in S\] \[\sum_{x\in S}p_X\left(x\right)=1\]

The elements \(x\in S\) form a set partition of the state space \(S\); that is, all elements in \(S\) are disjoint. The probability \(\mathbb{P}_X\left(X\in A\right)\) where \(A\in\mathcal{X}\) and \(\mathcal{X}\) is the support of \(\mathbb{P}_X\) is constructed through the summation of the pmf:

\[\begin{align} \mathbb{P}_X\left(X\in A\right)&=\sum_{x\in A}\mathbb{P}_X\left(X=x\right)\\ &=\sum_{x\in A}p_X\left(x\right) \end{align}\]

The cumulative distribution function (cdf) \(F_X\left(x\right)\) of a discrete random variable \(X\) is:

\[F_X\left(x\right)=\mathbb{P}_X\left(X\leq x\right)\,\,\forall\,x\in\mathbb{R}\]

The probability \(\mathbb{P}_X\left(x\in A\right)\) where \(A\subseteq S\) and \(A\) is a semi-closed interval \(\left(a_1, a_2\right]\) where \(a_1<a_2\) is constructed through the difference of cdfs:

\[\begin{align} \mathbb{P}_X\left(x\in A\right)&=\mathbb{P}_X\left(X\leq a_2\right)-\mathbb{P}_X\left(x\leq a_2\right)\\ &=F_X\left(a_2\right)-F_X\left(a_1\right) \end{align}\]