Probability Mass Function

Used in the context of discrete Random Variable. The p.m.f. helps us understand how does the r.v. behave.

Probability Mass Function (Definition)

$f(x)$ is the p.m.f. of $X$ if $f(x)=P(X=x)$.

See PDF for Continuous Random Variable.

Ex: Roll a die twice. $X$ = sum of the faces. Find the p.m.f. of $X$:

• $f(1)=0$
• $f(2)=\frac{2}{36}$
• $f(2)=\frac{3}{36}$
• $⋮$
• $f(12)=\frac{1}{36}$

Properties

Suppose $X$ is an r.v. with p.m.f. $f$, then

1. $f(x)\ge 0\forall x$
2. ${\sum }_{x\in X}f(x)=1$

pmf to cdf:

• $F(x)={\sum }_{y:y\le x}f(y)$

Related

• See Distribution for conversions
• Cumulative Distribution Function
• Probability Density Function for continuous Probability Density Function for continuous Random Variable