Cumulative Distribution Function (CDF)

Study guides: Exam P-1, Exam MAS-I

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 and right-continuous
$\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$

CDF and Probability for a Uniform Variable If $X$ is uniform on $[0, 5]$ , what is the CDF and what is $P(2 \leq X \leq 4)$ ?

Answer The PDF is $f(x) = 1/5$ for $0 \leq x \leq 5$ , so the CDF is: $F(x) = \frac{x}{5}, \quad 0 \leq x \leq 5$ $P(2 \leq X \leq 4) = F(4) - F(2) = \frac{4}{5} - \frac{2}{5} = \frac{2}{5} = 0.4$