PMF is a function that maps possible outcomes of a discrete random variables to the corresponding actual probabilities. For random variable Y, we have:
\begin{equation} f(k) = P(Y=k) \end{equation}
and f is a function that is the PMF, which is the mapping between a random variable and a value it takes on to the probability that the random variable takes on that value. Shorthand \begin{equation} P(Y=k) = p(y), where\ y=k \end{equation} its written smaller y represents a case of Y where Y=y. Properties of PMD \begin{equation} 0 \leq P(x) \leq 1 \end{equation} and
\begin{equation} \sum_{}^{} P(x) = 1 \end{equation}