A continuous random variable can take uncountably infinitely many values in an interval. Measurements such as height, mass and temperature are commonly modelled as continuous. Normal random variables A normal random variable is continuous, with values concentrated around a central mean and symmetric tails for unusually small and unusually large values. If X N(mu,sigma2), where sigma>0, then E(X)=mu, Var(X)=sigma2 and SD(X)=sigma. Its mean, median and mode are all mu. Density and cumulative probability f(x)=(1)/((2pi))exp[-((x-mu)2)/(2sigma2)], F(x)=P(X <= x)= _-infinityxf(t),dt The value f(x) is a density, not P(X=x). For every particular value x, P(X=x)=0. Hence P(X <= x)=P(X<x) and P(X >= x)=P(X>x).

A continuous random variable can take uncountably infinitely many values in an interval. Measurements such as height, mass and temperature are commonly modelled as continuous.

A normal random variable is continuous, with values concentrated around a central mean and symmetric tails for unusually small and unusually large values.

If \(X\sim N(\mu,\sigma^2)\), where \(\sigma>0\), then \(E(X)=\mu\), \(\operatorname{Var}(X)=\sigma^2\) and \(\operatorname{SD}(X)=\sigma\). Its mean, median and mode are all \(\mu\).

Normal-distribution notation

\[f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right],\qquad F(x)=P(X\leq x)=\int_{-\infty}^{x}f(t)\,\mathrm{d}t\]

Enrichment — outside the assessed syllabus

The value \(f(x)\) is a density, not \(P(X=x)\). For every particular value \(x\), \(P(X=x)=0\). Hence \(P(X\leq x)=P(X<x)\) and \(P(X\geq x)=P(X>x)\).

Continuous and Normal Random Variables

Continuous and Normal Random Variables