Probability density function
Undergraduate
Consider a random variable $X$ with real values. A function $f \geq 0$ is a probability density of $X$ if probabilities are areas under the graph of $f$:
\[P(a \leq X \leq b) = \int_a^b f(x) \mathrm{d}x \quad \text{for all } a \leq b.\]Properties
- The integral of every probability density $f(x)$ over its domain is 1: \(\int_{-\infty}^\infty f(x) \mathrm{d}x = 1.\)
- A density can be greater than $1$: the uniform distribution on $[0, \frac{1}{2}]$ has density $2$. The value $f(x)$ is not a probability; only integrals of $f$ are.
- For a continuous random variable, any individual value has probability $0$.