Using a PDF: Probabilities, Mean, Variance and Median (9709 PS2 – 6.3)

Core Concept

A continuous random variable $X$ has PDF $f(x)$ if $f(x) \geq 0$ for all $x$ and the total area under the curve equals 1. All probabilities and summary measures follow from integration of $f(x)$ directly; the syllabus does not require you to write down a separate cumulative distribution function $F(x)$, though the underlying idea of accumulating area is exactly what you use.

Probability as area: $P(a < X < b)$ is the area under $f(x)$ between $a$ and $b$. Because $X$ is continuous, $P(X = a) = 0$, so strict and non-strict inequalities are interchangeable.

Mean: $\text{E}(X) = \int x\,f(x)\,dx$ over the support — this is the balance point of the distribution.

Variance: $\text{Var}(X) = \text{E}(X^2) - [\text{E}(X)]^2$, where $\text{E}(X^2) = \int x^2 f(x)\,dx$.

Median $m$: The value such that exactly half the total area lies to the left, i.e. the area from the lower bound of the support to $m$ equals $\tfrac{1}{2}$. More generally, the $p$th percentile $x_p$ satisfies: area from lower bound to $x_p$ equals $p$.