Probabilities as areas

The distributions of a continuous random variable is defined by a type of histogram called a probability density function (pdf), with the following properties

Based on the properties of histograms, the probability of a value between any two constants is the area under the pdf above this range of values.

Probabilities by integration

Since probability density functions can usually be expressed as simple mathematical functions, these areas can be found as integrals,

\[ P(a \lt X \lt b) \;\; = \; \; \int_a^b {f(x)}\; dx \]

Properties of a probability density function

A function \(f(x)\) can be the probability density function of a continuous random variable if and only if

\[ f(x) \;\; \ge \; \; 0 \quad\quad \text{for all } x \text{, and} \] \[ \int_{-\infty}^{\infty} {f(x)}\; dx \;\; = \; \; 1 \]

(Proved in full version)