Probability Density Functions of Continuous Random Variables

A continuous random variable $X$ can take any value in some interval of $\mathbb{R}.$ Unlike a discrete random variable, the probability that $X$ equals any single value is zero. Instead, the distribution of $X$ is described by a probability density function (PDF) $f(x),$ which must satisfy two properties:

1. $f(x) \geq 0$ for all $x \in \mathbb{R}.$
2. $\displaystyle\int_{-\infty}^{\infty} f(x)\, dx = 1.$

Property 1 ensures that densities are never negative; property 2 ensures that the total probability is $1.$

For instance, consider

$$f(x) = \begin{cases} 2x, & 0 \leq x \leq 1, \\ 0, & \text{otherwise.}\end{cases}$$

The function is non-negative on $[0,1],$ and

$$\int_{-\infty}^{\infty} f(x)\, dx = \int_0^1 2x\, dx = x^2 \Big|_0^1 = 1.$$

Both properties hold, so $f$ is a valid PDF.