Cumulative Distribution Function

Compact study note.

Summary

The cumulative distribution function gives probability that random variable values are at or below chosen thresholds. Every real-valued random variable has a CDF.^[1]

Prerequisites

• Random Variable

Notation and Assumptions

$F_X(x)=P(X\le x)$ for all real $x$ . CDFs are nondecreasing, right-continuous, and have limits $0$ at $-\mathrm{\infty }$ and $1$ at $+\mathrm{\infty }$ .

Essential Result

For continuous variables, interval probability is CDF difference:

For discrete variables, jump sizes give point probabilities.

Small Example

If $X$ is uniform on $[0,1]$ , then $F_X(0.3)=0.3$ .

Common Mistakes

• Assuming every CDF is continuous; discrete CDFs jump.
• Using $F_X(b)-F_X(a)$ with wrong endpoint conventions for atoms.

Connections

• Quantiles
• Probability Density Function
• Discrete Random Variable

References