Mathematics and Statistics · Ch 16 — Probability Distributions

Probability Density Function of a Continuous Random Variable (p.d.f.)

A continuous random variable takes uncountably many values, so we can no longer attach a positive probability to each single value — in fact for a continuous $X$ , $P(X = c) = 0$ for any particular number $c$ . Instead, probability is described by a probability density function (p.d.f.) $f(x)$ , and probabilities are obtained as areas under the curve $y = f(x)$ , i.e. by integration.

Note

Conditions for a valid p.d.f.

A function $f(x)$ is a probability density function of a continuous random variable $X$ iff:

$f(x) \ge 0$ for all $x$ , and
$\displaystyle\int_{-\infty}^{\infty} f(x)\,dx = 1$ (the total area under the curve is $1$ ).

The probability that $X$ lies between two values $a$ and $b$ is the area under $f$ over that interval:

P(a \le X \le b) = \int_{a}^{b} f(x)\,dx.

Because a single point contributes zero area, it makes no difference whether the endpoints are included:

P(a \le X \le b) = P(a < X < b) = P(a \le X < b) = P(a < X \le b).

This is a genuine change of viewpoint from the discrete case: a sum of probabilities becomes an integral of a density, and $f(x)$ itself is not a probability (it may even exceed $1$ ) — only the area it encloses is a probability. …

Definition 6Probability density function (p.d.f.)

For a continuous $X$ , a function $f(x)\ge 0$ with $\int_{-\infty}^{\infty} f(x)\,dx = 1$ ; probabilities are areas: $P(a\le X\le …

Definition 7Area-as-probability rule

For a continuous variable $P(X=c)=0$ , so interval probabilities are unaffected by whether endpoints are included; probability equal …

Start your 14-day free trial to unlock the full solution →

← Previous · 2Probability Distribution of a Discrete Random Variable (p.m.f.)Next · 4 →Expected Value (Mean) and Variance