Continuous Probability Densities

A continuous random variable models a quantity such as height, mass or time. Its probability density function (p.d.f.) is written as \(\mathrm{f}(x)\).

Suppose \(\mathrm{f}(x)=0\) outside \(a\le x\le b\). The density satisfies

\(\mathrm{f}(x)\ge0\) for every \(x\).

\(\int_{-\infty}^{\infty}\mathrm{f}(x)\,\mathrm{d}x=\int_a^b\mathrm{f}(x)\,\mathrm{d}x=1\)

Probabilities as areas

For \(a\le x_1\le x_2\le b\),

\(\mathrm{P}(X\le x_1)=\int_a^{x_1}\mathrm{f}(x)\,\mathrm{d}x\)

\(\mathrm{P}(X\ge x_2)=\int_{x_2}^b\mathrm{f}(x)\,\mathrm{d}x\)

\(\mathrm{P}(x_1\le X\le x_2)=\int_{x_1}^{x_2}\mathrm{f}(x)\,\mathrm{d}x\)

Take note

At a single point, \(\mathrm{P}(X=c)=0\). Therefore including or excluding an interval endpoint does not change its probability:

\(\begin{aligned}\mathrm{P}(x_1\le X\le x_2)&=\mathrm{P}(X=x_1)+\mathrm{P}(x_1<X<x_2)+\mathrm{P}(X=x_2)\\&=0+\mathrm{P}(x_1<X<x_2)+0\\&=\mathrm{P}(x_1<X<x_2).\end{aligned}\)

The probabilities \(\mathrm{P}(x_1<X\le x_2)\) and \(\mathrm{P}(x_1\le X<x_2)\) have this same value.