The Continuous Uniform Distribution

This lesson introduces the continuous uniform distribution on an interval $[a,b]$ . We define its probability density function, compute probabilities of subintervals, derive formulas for the mean and variance, and solve for thresholds given a target probability.

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Tutorial

The Uniform Density

A continuous random variable $X$ has a continuous uniform distribution on the interval $[a,b]$ when it is equally likely to take any value in that interval. We write $X \sim U(a,b)$ .

The probability density function (PDF) is constant on $[a,b]$ and zero elsewhere:

f(x) = \begin{cases} \dfrac{1}{b-a} & \text{if } a \le x \le b \\[3pt] 0 & \text{otherwise} \end{cases}

The height $\dfrac{1}{b-a}$ is exactly what's needed so that the total area under the PDF equals $1$ .

For any subinterval $[c,d] \subseteq [a,b]$ , the probability is the area of a rectangle:

P(c \le X \le d) = \int_c^d \frac{1}{b-a}\, dx = \frac{d-c}{b-a}.

Quick example. If $X \sim U(0,10)$ , then $f(x) = \dfrac{1}{10}$ on $[0,10]$ and

P(2 \le X \le 5) = \frac{5-2}{10-0} = \frac{3}{10}.