![Uniform distribution with constant pdf on [a,b], showing cdf, mean and area interpretation](/slides/probability/1.5.common-distributions/1.5.5.uniform-distribution.png)
What is a Uniform Distribution?
The standard uniform distribution $U$ is a continuous random variable which is equally likely to be in any interval of the same width within the set $[0,1]$.
A general uniform distribution $X$ is equally likely to be in any interval of the same width within a set $[a,b]$ for two fixed real numbers $b>a$.
It is sometimes called the “box” distribution, as its PDF looks like a box.
| Notation: | $X \sim \mathcal{U}[a,b]$ |
| Type: | $ \text{Continuous} $ |
| PDF: | $ f_X(x) = \frac{1}{b-a}, \ \ a \le x \le b $ |
| CDF: | $ F_X(x) = \frac{x-a}{b-a}, \ \ a \le x \le b $ |
| Support: | $x \in [a,b] $ |
| Mean: | $\mathbb{E}(X) = \frac{a+b}{2}$ |
| Variance: | $\mathbb{V}\text{ar}(X) = \frac{(b-a)^{2}}{12} $ |
This should not be confused with the discrete uniform distribution, a discrete random variable equally like to take any value within the set $ \{ 1,2,\dots,n \} $. This models rolling a fair die.