## Definition The ==Cumulative Distribution Function (CDF)== of a random variable $X$ gives the probability that $X$ takes a value less than or equal to $x$: $ F(x) = P(X \leq x) $ For a continuous random variable, $F(x) = \int_{-\infty}^{x} f(t)\, dt$. The CDF is non-decreasing, right-continuous, and satisfies $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$. > [!example]- If $X$ is uniform on $[0, 5]$, what is the CDF and what is $P(2 \leq X \leq 4)$? > The PDF is $f(x) = 1/5$ for $0 \leq x \leq 5$, so the CDF is: > $ F(x) = \frac{x}{5}, \quad 0 \leq x \leq 5 $ > $ P(2 \leq X \leq 4) = F(4) - F(2) = \frac{4}{5} - \frac{2}{5} = \frac{2}{5} = 0.4 $