What Is a Probability Density Function?

A probability density function (PDF) is a non-negative function that describes how probability is distributed across the possible values of a continuous random variable. Probability is represented by area under the density curve, not by the function's height at one point.

The Reasoning

If a continuous random variable $X$ has probability density function $f$, then the probability that $X$ lies between $a$ and $b$ is

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

A valid PDF must satisfy both conditions:

The function's height indicates relative concentration, so larger values of $f$ suggest that nearby outcomes are more densely distributed. However, only an integral over an interval produces a probability.

A useful related function is the cumulative distribution function, $F(x)=P(X\leq x)$, which equals $\int_{-\infty}^{x}f(t)\,dt$. Thus, is non-decreasing and, where differentiable, .

In piecewise exam questions, first identify the support: the interval on which $f$ is non-zero. Then use the total-area condition to determine any unknown constant before calculating the requested probability. Keep bounds consistent with the stated support to avoid integrating an invalid formula.

For example, suppose $f(x)=2x$ for $0 \leq x \leq 1$, and $f(x)=0$ otherwise. It is valid because and

$$\int_{0}^{1}2x\,dx=\left[x^2\right]_{0}^{1}=1.$$

Therefore,

$$P(0.5\leq X\leq1)=\int_{0.5}^{1}2x\,dx =\left[x^2\right]_{0.5}^{1}=0.75.$$

For a continuous variable, $P(X=a)=0$. Consequently, including or excluding interval endpoints does not change the probability: $P(a<X<b)=P(a\leq X\leq b)$.

A common misconception is that $f(x)$ itself is a probability and therefore cannot exceed $1$. This is false: a density may be greater than $1$, provided it remains non-negative and its total area is $1$.

IB Exam Technique

This is IB Mathematics: Analysis and Approaches HL, sub-topic AHL 4.14: Properties of discrete and continuous random variables. On Paper 1, show the integral setup and exact evaluation; on Papers 2 or 3, write the required integral before using the GDC. Examiners may ask you to determine an unknown parameter, calculate an interval probability, or find the median using $\int_{-\infty}^{m}f(x)\,dx=\frac{1}{2}$.