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Uniform Distribution

Exhibit 33.15 Uniform probability distribution.

Uniform distribution, or rectangular distribution, is a distribution that has constant probability over a fixed range. A random variable, X, is uniform on [a,b] if X is equally likely to take any value in the range from a to b (Exhibit 33.15).

Notation: [a, b] means continuous values from a to b.

$$X∼U[a,b]$$ $$f(t)=\begin{cases}\frac{1}{b-a}, \,\,if \,\,a \, \le t \le b \\ \\0 \,otherwise \end{cases}$$

The cumulative distribution function (cdf):

$$F(t)=\begin{cases}0 \,\,if \,\,t< a \\ \\ \frac{t-a}{b-a} \,\,if \,\,a \le t \le b \\ \\ 1 \,\,if \,\,t>b \end{cases}$$ $$E(X)= \frac{b-a}{2}, \,Var(X)=\frac{b-a}{12}$$

MarketingMind’s content is sourced from the Marketing Analytics Practitioner’s Guide (MAPG), a comprehensive 4-volume compendium on Marketing, Marketing Analytic and Market Research available in both physical and eGuide formats. (Click to learn more).

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Exhibit 33.15 Uniform probability distribution.

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