PDFs and CDFs — Section 4: Random Variables & Distributions

Every random variable, discrete or continuous, has a cumulative distribution function (CDF):

F(x) = P(X \le x)

It's non-decreasing, right-continuous, $F(-\infty) = 0$ , and $F(\infty) = 1$ . Because the CDF is universal, it's the cleanest way to compare or describe distributions.

For continuous variables, the probability density function (PDF) is the derivative of the CDF:

f(x) = F'(x), \qquad P(a \le X \le b) = \int_a^b f(x)\, dx = F(b) - F(a)

The PDF doesn't give probability at a single point — $f(x)$ is a density, not a probability. It can exceed $1$ (e.g. a Uniform on $[0, 0.5]$ has $f = 2$ on its support).

For discrete variables, the PMF and CDF are linked by

F(x) = \sum_{y \le x} p(y)

CDFs are easier to plot and compare, and they handle mixed distributions cleanly. PDFs and PMFs are easier for computing expectations and conditional distributions. Most calculations bounce between the two representations as needed.

Every random variable, discrete or continuous, has a cumulative distribution function (CDF):

F(x) = P(X \le x)

It's non-decreasing, right-continuous, $F(-\infty) = 0$ , and $F(\infty) = 1$ . Because the CDF is universal, it's the cleanest way to compare or describe distributions.

For continuous variables, the probability density function (PDF) is the derivative of the CDF:

f(x) = F'(x), \qquad P(a \le X \le b) = \int_a^b f(x)\, dx = F(b) - F(a)

The PDF doesn't give probability at a single point — $f(x)$ is a density, not a probability. It can exceed $1$ (e.g. a Uniform on $[0, 0.5]$ has $f = 2$ on its support).

For discrete variables, the PMF and CDF are linked by

F(x) = \sum_{y \le x} p(y)