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 $F$ , the pmf is recovered as the jump size of $F$ at each of its points of discontinuity.

Working rule. If $X$ takes the values $x_1<x_2<x_3<\cdots$ with cdf $F(x_i)$ , then

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

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

Note

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

Figure 11.9Cumulative distribution function $F(x)$ (step function) for the total score in two throws: $0$ for $x<2$, $\frac{1}{36}$ for $2\le x<3$, $\frac{5}{36}$ for $3\le x<4$, $\frac{15}{36}$ for $4\le x<5$, $\frac{27}{36}$ for $5\le x<6$, $1$ for $x\ge 6$ (Fig. 11.9)

$Fig. 11.9 — Cumulative distribution function $F(x)$ (step function) for the total score in two throws: $0$ for $x<2$, $\frac{1}{36}$ for $2\le x<3$, $\frac{5}{36}$ for $3\le x<4$, $\frac{15}{36}$ for $4\le x<5$, $\frac{27}{36}$ for $5\le x<6$, $1$ for $x\ge 6$ (Fig. 11.9)$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Cumulative distribution function $F(x)$ (step function) for the total score in two throws: $0$ for $x<2$ , $\frac{1}{36}$ for $2\le x<3$ , $\frac{5}{36}$ for $3\le x<4$ , $\frac{15}{36}$ for $4\le x<5$ , $\frac{27}{36}$ for $5\le x<6$ , …

Figure 11.10Distribution function F(x) of a discrete random variable drawn as a right-continuous step function, with jumps 0.25, 0.35, 0.30, 0.10 at x = -2, -1, 0, 1 giving cumulative heights 0.25, 0.60, 0.90 and 1

Fig. 11.10 — Distribution function F(x) of a discrete random variable drawn as a right-continuous step function, with jumps 0.25, 0.35, 0.30, 0.10 at x = -2, -1, 0, 1 giving cumulative heights 0.25, 0.60, 0.90 and 1

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Distribution function F(x) of a discrete random variable drawn as a right-continuous step function, with jumps 0.25, 0.35, 0.30, 0.10 at x = -2, -1, 0, 1 giving cumulative heights 0. …

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