Uniform distribution
Definition taught in Random variables and distributions, part of Probability
Uniform distribution
| Type | Definition |
| Taught in | Random variables and distributions |
| Part | Probability |
| Properties | 1 |
| Worked examples | 1 |
| Questions using it | 5 |
X is uniform on the interval [a,b], written X\sim\operatorname{U}[a,b], when its density is 1/(b-a) on that interval and zero elsewhere. Then \mathbb{E} X=(a+b)/2 and \operatorname{Var}(X)=(b-a)^2/12.
Properties
Inverse-transform method
Main article: Inverse-transform method
Let F be a distribution function and define F^{-1}(u)=\inf\{x:F(x)\ge u\} for u\in(0,1). If U\sim\operatorname{U}[0,1], then X=F^{-1}(U) has distribution function F.
Examples
The sum of two uniforms
Worked example from Functions of random variables and limit theorems
Let X and Y be independent and uniform on [0,1], and Z=X+Y.
Question. Find the density of Z.
Solution. By the convolution formula, f_Z(z)=\int f_X(x)f_Y(z-x)\,\mathrm{d} x. The integrand is 1 when both 0\le x\le1 and 0\le z-x\le1, that is when \max(0,z-1)\le x\le\min(1,z), and 0 otherwise. The length of that interval is z for 0\le z\le1 and 2-z for 1\le z\le2, so
f_Z(z)=\begin{cases}z,&0\le z\le1,\\ 2-z,&1\le z\le2.\end{cases}
The sum of two flat densities is a triangle. Adding more uniforms rounds the shape off further, towards the normal curve of the central limit theorem below.
Used in
- Taught in Random variables and distributions, where the derivation and examples are.
- Also used in Poisson processes · Estimation
- Questions that use it (5) — Draws from a uniform interval · Minimum and average of uniform draws · State the probability functions · The sum of uniform draws · A doubly stochastic matrix
- All 27 questions on Random variables and distributions
See also
Random variable · Cumulative distribution function · Discrete distribution · Continuous distribution · Expectation · Expectation of a function · Variance · Probability generating function