Uniform distribution

A uniformly distributed random variable $𝑋$ represents the selection of a real number completely randomly over a given interval $[𝑎, 𝑏]$ . It is the simplest continuous distribution.

𝑋 \sim U (𝑎, 𝑏)

It has the following probability density function and Cumulative distribution function

𝑓 𝑋 (𝑥) = {1 𝑏 - 𝑎 𝑎 \leq 𝑥 \leq 𝑏 0 o t h e r w i s e 𝐹 𝑋 (𝑥) = ⎧ { {⎨ { {⎩ 𝑥 - 𝑎 𝑏 - 𝑎 𝑎 \leq 𝑥 \leq 𝑏 0 𝑥 < 𝑎 1 𝑥 > 𝑏

Properties

Let $𝑋 \sim U (𝑎, 𝑏)$ .

Expectation: $𝜇 = 𝔼 [𝑋] = 𝑏 - 𝑎 2$
Variance: $𝜎 2 = V a r [𝑋] = (𝑏 - 𝑎) 2 12$
Moment-generating function: $𝑀 𝑋 : ℝ \to ℝ : 𝑡 \mapsto e 𝑡 𝑏 - e 𝑡 𝑎 𝑡 (𝑏 - 𝑎)$

Standard form

The standard uniform distribution $𝑋 \sim U (0, 1)$ is particularly useful, as it is a natural starting point for transforming random variables. See Universality of the uniform distribution

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