A waiting time or a measured length can take any value in an interval, so listing $P(X = x)$ value by value no longer works. In continuous random variables probability became area under a curve. With the Fundamental Theorem of Calculus we can compute those areas exactly.
Key idea
A probability density function $f$ satisfies $f(x) \ge 0$ and $\int_{-\infty}^{\infty} f(x)\,dx = 1$, and $$P(a \le X \le b) = \int_{a}^{b} f(x)\,dx.$$ A single point has probability $P(X = a) = 0$, so $<$ and $\le$ give the same answers. The height $f(x)$ is not a probability and may be larger than 1.
Why it works
For a short interval near $x$, $P(x \le X \le x + \Delta x) \approx f(x)\,\Delta x$. So $f(x)$ is probability per unit length, just as a density in physics is mass per unit volume. A probability per unit can be large when the unit is long compared with where the probability sits: if all the probability is packed into $[0, \frac{1}{4}]$, the density there must be 4 to make the area 1. And a single point is an interval of length 0, so it carries no area: $P(X = a) = \int_{a}^{a} f = 0$.
Worked examples
Worked example: Finding the constant
$f(x) = cx^{2}$ for $0 \le x \le 2$ and 0 otherwise. Find $c$.
- The total area must be 1: $\int_{0}^{2} cx^{2}\,dx = c \cdot \frac{8}{3}$.
- $c \cdot \frac{8}{3} = 1$, so $c = \frac{3}{8}$.
Answer: $c = \frac{3}{8}$
Worked example: A probability as an area
With $f(x) = \frac{3}{8}x^{2}$ on $[0, 2]$, find $P(1 \le X \le 2)$ and $P(X > 3)$.
- $$P(1 \le X \le 2) = \int_{1}^{2} \frac{3}{8}x^{2}\,dx = \frac{3}{8} \cdot \frac{8 - 1}{3} = \frac{7}{8}$$
- $P(X > 3) = 0$: the density is 0 beyond 2.
- Notice that $f(2) = \frac{3}{2}$, more than 1, and nothing is wrong.
Answer: $\frac{7}{8}$ and 0
Worked example: A tall density
$X$ is uniform on $\left[0, \frac{1}{4}\right]$, so $f(x) = 4$ there. Find $P(X \le 0.1)$ and $P(X = 0.1)$.
- $P(X \le 0.1) = 4 \times 0.1 = 0.4$: an area, height times width.
- $P(X = 0.1) = 0$: a single point has no width.
Answer: $0.4$ and $0$
Common mistake
Reading $f(x)$ as the probability that $X = x$. A density can be 4, or 100, at a point, while $P(X = x) = 0$ for every single value. Check: a probability is always an area under $f$ over an interval, and must end up between 0 and 1.