PDF & CDF

The mathematics of uncertainty

For continuous quantities like a height, a weight, or a pixel intensity, asking P(X = 3.0000…) is hopeless: there are infinitely many values, so any single one has probability zero. Instead we describe how probability is spread with a probability density function f(x), and read off probabilities as areas.

A density isn't a probability itself, and it can exceed 1. What must hold is that it's non-negative and the total area is 1, the continuous echo of "the PMF sums to 1":

Drag μ and σ above: the curve slides and stretches, but the area underneath always stays exactly 1. Probability of an interval is the slice of area sitting over it.

Where this lives in MLA generative model's output p(x | θ) is a density. To sample from a 1-D distribution you can use inverse-transform sampling: draw a uniform u ∈ [0,1] and return F⁻¹(u), inverting the CDF. Normalizing flows generalize exactly this idea, learning an invertible map whose change-of-variables turns a simple density into a complex one.

▶ PDF & CDF

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