Mathematics · Ch 11 — Probability Distributions

Distribution function (Cumulative distribution function)

11.4.3

Distribution function (Cumulative distribution function)

Definition 11.7 (Cumulative distribution function, continuous case). For a continuous random variable $X$ with pdf $f(x)$ , the distribution function is

F(x)=P(X\le x)=\int_{-\infty}^{x} f(u)\,du,\qquad -\infty<x<\infty.

Remarks (comparing the two cases).

(1) In the discrete case $f(a)=P(X=a)$ directly; in the continuous case $f(a)$ is not the probability that $X=a$ — indeed $P(X=a)=0$ for every $a$ , by Definition 11.5.

(2) Passing from discrete to continuous simply replaces every sum by the corresponding integral.

(3) Because a single point carries no probability, the four inequality versions of an interval event all coincide for a continuous $X$ : $P(a\le X\le b)=P(a<X\le b)=P(a\le X<b)=P(a<X<b)$ — the endpoints can be included or excluded freely. …

Figure 11.12Probability density function f(x) = (1/21)x^2 on the interval 1 < x < 4, with the entire region under the curve shaded to show the total area equals 1

Fig. 11.12 — Probability density function f(x) = (1/21)x^2 on the interval 1 < x < 4, with the entire region under the curve shaded to show the total area equals 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. Probability density function f(x) = (1/21)x^2 on the interval 1 < x < 4, with the entire region under the curve shaded to show the total a …

Figure 11.13Probability density function f(x) = (1/21)x^2 with the area under the curve between x = 1.5 and x = 3.5 shaded, giving P(1.5 < X < 3.5) = 79/126

Fig. 11.13 — Probability density function f(x) = (1/21)x^2 with the area under the curve between x = 1.5 and x = 3.5 shaded, giving P(1.5 < X < 3.5) = 79/126

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. Probability density function f(x) = (1/21)x^2 with the area under the curve between x = 1.5 and x = 3.5 shaded, giving P(1.5 < X < 3. …

Figure 11.14Probability density function f(x) = (1/21)x^2 with the area under the curve between x = 1 and x = 2 shaded, giving P(X <= 2) = 7/63

Fig. 11.14 — Probability density function f(x) = (1/21)x^2 with the area under the curve between x = 1 and x = 2 shaded, giving P(X <= 2) = 7/63

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. Probability density function f(x) = (1/21)x^2 with the area under the curve between x = 1 and x = 2 shaded, giving P(X <= …

Figure 11.15Probability density function f(x) = (1/21)x^2 with the area under the curve between x = 3 and x = 4 shaded, giving P(3 < X) = 37/63

Fig. 11.15 — Probability density function f(x) = (1/21)x^2 with the area under the curve between x = 3 and x = 4 shaded, giving P(3 < X) = 37/63

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. Probability density function f(x) = (1/21)x^2 with the area under the curve between x = 3 and x = 4 shaded, giving P(3 < X …

Start your 14-day free trial to unlock the full solution →

← Previous · 11.4.2Probability density function Next · 11.4.3.1 →Properties of distribution function