Probability Theory 12 | Cumulative Distribution Function [dark version]

Every real-valued random variable comes with a cumulative distribution function (CDF) that turns probability questions into a single, monotone curve on the real line. The CDF, usually written as F_X(x), is defined by F_X(x)=P(X≤x): it accumulates all probability mass from −∞ up to the threshold x. That “accumulation” is why the function is called cumulative, and why it works the same way whether X is discrete (only certain values matter) or absolutely continuous (probability comes from a density).

The CDF has three key properties that follow directly from the rules of probability. First, as x→−∞, the interval (−∞,x] shrinks to the empty set, so F_X(x)→0. Second, as x→+∞, the interval expands to the entire real line, so F_X(x)→1. Third, F_X(x) is monotone increasing: if x1<x2, then (−∞,x1] is a subset of (−∞,x2], and probabilities can only stay the same or increase. Unlike a probability density function (PDF), which can rise and fall, the CDF never decreases.

A subtle but important detail is right-continuity. CDFs may have jumps, especially for discrete random variables. Right-continuity means the value at x0 matches the limit taken from values slightly larger than x0. Graphically, at a jump point, the filled-in point sits at the upper side of the jump because the event X=x0 contributes nonzero probability, and that mass is included when computing P(X≤x) for thresholds at x0.

The video then grounds the theory in the normal distribution, the central example for CDFs. For the standard normal case with mean μ=0 and standard deviation σ=1, the PDF is the Gaussian bell curve: f(x)= (1/√(2π))·e^(−x^2/2). When the distribution is absolutely continuous, the CDF is obtained by integrating the PDF from −∞ to x: F(x)=∫_{−∞}^{x} f(t) dt. Symmetry of the normal density implies F(0)=1/2, since half the probability lies to the left of zero.

To make these functions concrete, the transcript describes plotting both the PDF and CDF over a grid (for example, from −10 to 10 with a small step size). The PDF appears as a bell curve peaking at x=0, while the CDF rises smoothly from 0 toward 1 as x increases. Finally, it connects the theory to simulation: using R’s normal generator (rnorm) to draw many samples (e.g., 6000) and comparing the resulting histogram to the bell curve. The takeaway is that the CDF provides a unified, threshold-based probability function, while the PDF supplies the density that—when integrated—produces that cumulative curve.