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Mathematics · Ch 11 — Probability Distributions

Probability Mass Function from Cumulative Distribution Function

11.3.5

Probability Mass Function from Cumulative Distribution Function

The reverse conversion is just as direct: given the cdf FF, the pmf is recovered as the jump size of FF at each of its points of discontinuity.

Working rule. If XX takes the values x1<x2<x3<⋯x_1<x_2<x_3<\cdots with cdf F(xi)F(x_i), then

f(xi)=F(xi)−F(xi−1),i=1,2,3,…f(x_i)=F(x_i)-F(x_{i-1}),\qquad i=1,2,3,\dots

(with the convention F(x0)=0F(x_0)=0, i.e. the value of FF just before the first jump).

Note

FF is non-decreasing and right-continuous, so its left-hand limit F(a−)F(a^-) always exists, and the jump of FF at a point aa is F(a)−F(a−)F(a)-F(a^-). This jump is precisely P(X=a)=F(a)−F(a−)P(X=a)=F(a)-F(a^-) — it is the probability mass sitting at aa. The set of discontinuities of a cdf is therefore at most countable, matching the countable support of the discrete random variable it belongs to. …

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