Probability Density Functions — CAIE A-Level Mathematics

Introduction

A continuous random variable (CRV) can take any value in an interval — for example, the exact height of a randomly chosen student, or the time until a component fails. Unlike a discrete random variable, we cannot list individual probabilities for each outcome; instead, probability is described by a probability density function (PDF), which spreads probability continuously across an interval.

In the 9709 Paper 6 exam, questions on PDFs regularly ask you to verify that a function is a valid PDF, find unknown constants, and calculate probabilities as definite integrals. Mastering this topic is therefore essential for confident performance in PS2.

Core Concept

Continuous Random Variables

A continuous random variable $X$ takes values over an interval (which may be infinite). Individual point probabilities are always zero: $P(X = a) = 0$ for any specific value $a$ . Probability is only meaningful over an interval.

The Probability Density Function

A function $f(x)$ is a valid probability density function for a CRV $X$ if and only if it satisfies two conditions:

Non-negativity: $f(x) \geq 0$ for all $x$ in the domain (and $f(x) = 0$ outside the domain).
Total area equals 1: The integral of $f(x)$ over the entire domain equals 1.

Probabilities are computed as areas under the curve:

P(a \leq X \leq b) = \int_a^b f(x)\, dx

Because $P(X = a) = 0$ , it makes no difference whether the endpoints are included:

P(a \leq X \leq b) = P(a < X < b) = P(a \leq X < b) = P(a < X \leq b)

Infinite Domains

The syllabus explicitly includes PDFs defined on infinite intervals, such as $f(x) = \dfrac{4}{x^3}$ for $x \geq 1$ . In these cases, the total-area condition involves an improper integral:

\int_1^{\infty} \frac{4}{x^3}\, dx = \lim_{t \to \infty} \int_1^{t} 4x^{-3}\, dx

provided the limit exists and equals 1.

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