Density and Mass Functions
- Statistics 1.5

in Mathematics on Statistics · 1 min

Prabability Mass Function (PMF)

The probability mass function (PMF) of a discrete random variable $X$ is given by: \[ f_X(x) = P(X = x) \]

Probability Density Function (PDF)

The probability density function (PDF) of a continuous random variable $X$ is the function that satisfies: \[ F_X(x) = \int_{-\infty}^x f_X(t) \dd{t} \] for all $x\in\mathbb{R}$, where $F_X(x)$ is the cumulative distribution function (CDF) of $X$.

Properties

A function $f_X(x)$ is a PDF or a PMF of a random variable $X$ if and only if

  • $f_X(x) \geq 0$
  • $\sum_{x} f_X(x) = 1$ (for PMF)
  • $\int_{-\infty}^{\infty} f_X(x) \dd{x} = 1$ (for PDF)