Cumulative Distribution Functions for Continuous Random Variables

Defines the cumulative distribution function (CDF) $F(x) = P(X \le x)$ for a continuous random variable, computes the CDF from a PDF via integration, computes interval probabilities and tail probabilities using $F(b)-F(a)$ and $1-F(a)$ , recovers the PDF from a given CDF by differentiation, and solves for unknown constants in a CDF using the boundary condition $F(\text{upper}) = 1$ .

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Tutorial

Defining the CDF of a Continuous Random Variable

The cumulative distribution function (CDF) of a continuous random variable $X$ with probability density function $f$ is the function $F$ defined by

F(x) = P(X \le x) = \int_{-\infty}^{x} f(t)\,dt.

In words, $F(x)$ is the probability that $X$ takes a value at most $x$ . As $x$ sweeps from left to right, $F(x)$ accumulates area under the PDF.

For example, suppose $X$ has PDF

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

For $0 \le x \le 1,$ we compute

F(x) = \int_0^x 2t\,dt = t^2 \Big|_0^x = x^2.

Outside the support of $X,$ $F$ is constant: $F(x) = 0$ for $x < 0$ and $F(x) = 1$ for $x > 1.$

Every CDF satisfies $\lim_{x \to -\infty} F(x) = 0,$ $\lim_{x \to +\infty} F(x) = 1,$ and is non-decreasing.