2.2b.1 Probability Mass Function

Probability Mass Function (PMF)

The probability distribution of a discrete random variable $X$ is fully described by a non-negative probability mass function (PMF) $p_X:\mathcal{X}\to [0,1]$, formally defined as
$p_X(x)=\mathbb{P}[X=x],$
where $\{X=x\}\in \mathcal{F}$ is the event consisting of all outcomes where the random variable $X$ equals the specific value $x\in \mathcal{X}$.

The cumulative probability that $X$ takes on a value less than or equal to $x$ in the PMF is given by the monotonically non-decreasing cumulative distribution function (CDF) $F_X:\mathbb{R}\to [0,1]$, formally defined as
$F_X(x)=\mathbb{P}[X\le x]={\sum }_{k\le x}^{}{p}_X(k),$
implying that the PDF is the rate of change of the CDF, $p_X(x)=F_X(x)-F_X(x-1)$.

PMFs Sum to One ${\sum }_{x\in \mathcal{X}}^{}{p}_X(x)=1.$

For a PMF to meet the sum-to-one criterion of any probability function,

Properties

(Joint Distribution) $p_{X,Y}(x,y)=\mathbb{P}[X=x\cap Y=y]$.

(Conditionality) $p_{X\mid Y}(x\mid y)=\frac{p_{X,Y}(x,y)}{p_Y(y)}$.

(Marginality) $p_X(x)=\sum _{y\in \mathcal{Y}}{p}_{X,Y}(x,y)$.

(Independence) If $X,Y$ are independent, then $p_{X,Y}(x,y)=p_X(x)\cdot p_Y(y)$ and $p_{X\mid Y}(x\mid y)=p_X(x)$.