Probability density function (PDF) of a continuous random variable gives the relative probability for each of its possible values. Use this tag for discrete probability mass functions (PMFs) too.

Tag Usage

Use this tag when asking about probability functions in general, whether Probability density functions, or discrete probability mass functions (PMFs).

Overview

PDF stands for Probability Density Function; distinguished from CDF for Cumulative Distribution Function. A PDF describes the relative probability of a continuous random variable taking a given value. PMF stands for Probability Mass Function; it describes the probability of a discrete random variable taking a given value.

In case of continuous variables $X$, the PDF $\mathcal{P}_X(x)$ can be integrated over an interval $\mathcal{I}$ (or, more generally, any Borel set) to find the probability that the variable is in that interval:

$$\Pr(X \in \mathcal{I}) = \int_\mathcal{I} \mathcal{P}_X(x) dx.$$

Some common PDFs:

  • Normal: $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right]$

  • Gamma: $f(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}\exp(-\beta x)$

Some common PMFs:

  • Binomial: $\Pr(X = x) = \frac{n!}{x!(n-x)!}p^x(1-p)^{n-x}$ for integral $n\ge 0.$

  • Bernoulli: $\Pr(X = x) = p^x (1-p)^{1-x}$ for $x\in \{ 0,1 \}.$

References