A continuous random variable models a quantity such as height, mass or time. Its probability density function (p.d.f.) is written as f(x). Suppose f(x)=0 outside a <= x <= b. The density satisfies f(x) 0 for every x. _-infinityinfinityf(x),dx= _abf(x),dx=1 Probabilities as areas For a <= x_1 <= x_2 <= b, P(X <= x_1)= _ax_1f(x),dx P(X >= x_2)= _x_2bf(x),dx P(x_1 <= X <= x_2)= _x_1x_2f(x),dx Take note At a single point, P(X=c)=0. Therefore including or excluding an interval endpoint does not change its probability: P(x_1 <= X <= x_2)=P(X=x_1)+P(x_1<X<x_2)+P(X=x_2) =0+P(x_1<X<x_2)+0 =P(x_1<X<x_2). The probabilities P(x_1<X <= x_2) and P(x_1 <= X<x_2) have this same value.

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

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

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

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

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

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.

Continuous Random Variables

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Continuous Probability Densities

Continuous Probability Densities