Mathematics · Ch 11 — Probability Distributions

Probability density function from Probability distribution function

11.4.5

Probability density function from Probability distribution function

Working rule. If $F(x)$ is the cdf of a continuous random variable $X$ , its pdf is recovered by differentiation:

f(x)=\frac{dF(x)}{dx}=F'(x),\quad\text{wherever the derivative exists.}

At the finitely many points where two pieces of $F$ meet and the derivative is not defined (a "corner" of $F$ ), $f$ may be assigned any convenient value there without affecting any probability — since a single point never carries probability for a continuous $X$ (§11.4.1), the choice at that isolated point is immaterial to every computation. …

Figure 11.17Cumulative distribution function F(x) of the triangular density: F = 0 for x < 1, (x-1)^2/2 on 1 <= x < 2, 1 - (3-x)^2/2 on 2 <= x < 3, and 1 for x >= 3, drawn as a smooth S-shaped curve through (2, 0.5)

Fig. 11.17 — Cumulative distribution function F(x) of the triangular density: F = 0 for x < 1, (x-1)^2/2 on 1 <= x < 2, 1 - (3-x)^2/2 on 2 <= x < 3, and 1 for x >= 3, drawn as a smooth S-shaped curve through (2, 0.5)

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) of the triangular density: F = 0 for x < 1, (x-1)^2/2 on 1 <= x < 2, 1 - (3-x)^2/2 on 2 <= x < 3, and 1 for x >= 3, drawn as a smooth S-shaped cu …

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