Probability Mass Function (PMF)

A Probability Mass Function (PMF) gives the probability that a discrete Random Variable $X$ equals a specific value $x$ .

$p(x) = P(X = x)$

It must satisfy $p(x) \geq 0$ for all $x$ and $\sum_{\text{all } x} p(x) = 1$

PMF of a Fair Die A fair die is rolled. What is the PMF of the outcome $X$ ?

Answer Each face is equally likely, so the PMF is: $p(x) = \frac{1}{6}, \quad x \in \{1, 2, 3, 4, 5, 6\}$ We can verify: $\sum_{x=1}^{6} \frac{1}{6} = 1$ .