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Uniform Distribution

Also Uniform PDF · Continuous Uniform Distribution · Gleichverteilung · Stetige Gleichverteilung

The uniform distribution on an interval [a, b] has the constant density 1/(b − a) on [a, b] and zero elsewhere; its mean is (a + b)/2 and its variance (b − a)²/12.

Know these firstProbability Density Function Expected Value Variance

Formula#

\mu_X = \frac{a+b}{2}, \qquad \sigma_X^2 = \frac{(b-a)^2}{12}.

ProofDerivation of mean and variance

Normalization: $\int_a^b \frac{1}{b-a}\,\mathrm{d}x = 1$ .

Mean: $\displaystyle\int_a^b \frac{1}{b-a}\, x\,\mathrm{d}x = \frac{b^2 - a^2}{2(b-a)} = \frac{a+b}{2}$ .

Variance: $\sigma_X^2 = \int_a^b \frac{1}{b-a}(x - \mu_X)^2\,\mathrm{d}x$ . Substitute $y = x - \mu_X$ and write $c = b - a$ , so $y$ runs over $[-c/2, c/2]$ :

\sigma_X^2 = \frac1c \int_{-c/2}^{c/2} y^2\,\mathrm{d}y = \frac1c\left[\frac13\left(\frac c2\right)^3 - \frac13\left(-\frac c2\right)^3\right] = \frac{c^2}{12} = \frac{(b-a)^2}{12}.

Properties#

The variance depends only on the interval length $b - a$ , not on its position. The standard deviation is $\sigma_X = (b-a)/\sqrt{12}$ .
Every subinterval of the same length has the same probability: $P(X \in [x, x + h]) = h/(b-a)$ for $[x, x+h] \subseteq [a, b]$ .

Example#

Sources#

MSI script p. 15 (§2.2.2), MSI Slides 02 > 2.2.2 Simple Example: Uniform Distribution (not published)

Connections

Mentioned in inline links5

VL02 – Probability and Statistics in a NutshellMSI
For the Uniform Distribution on $[a,b]$ , integrating the PDF gives the mean $(a+b)/2$ and the variance $(b-a)^2/12$ (Script p. 15). §
Uniform Distribution: first worked PDF example §
Expected Value
For $X$ uniform on $[a, b]$ : $\mu_X = \int_a^b \frac{x}{b-a}\,\mathrm{d}x = \frac{a+b}{2}$ (Uniform Distribution). With $f(X) = 3 + 2X$ on $[0, 1]$ : $\mathbb{E}\{3 + 2X\} = 3 + 2 \cdot \tfrac12 = 4$ , but… §
Probability Density Function
The uniform distribution on $[0, \tfrac12]$ has $p_X(x) = 2$ on that interval and $0$ elsewhere. It integrates to $2 \cdot \tfrac12 = 1$ , and $P(X \in [0, \tfrac14]) = \int_0^{1/4} 2\,\mathrm{d}x = \tfrac12$ . §
Variance
For $X$ uniform on $[a, b]$ , $\sigma_X^2 = \frac{(b-a)^2}{12}$ (Uniform Distribution). Check with the formula on $[0, 1]$ : $\mathbb{E}\{X^2\} = \tfrac13$ , $\mu_X = \tfrac12$ , so $\sigma_X^2 = \tfrac13 - \tfrac14 = \tfrac1{12}$ . §
Random Variables and Distributions
A random variable models a quantity that is not known before the experiment, such as a noisy measurement. It is described by probabilities of events, or more conveniently by a PDF. Joint, marginal and conditional densities describe several… §
Uniform Distribution — the simplest PDF, constant on an interval §

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