22  Continuous Uniform Distribution

A continuous random variable \(X\) is said to have a uniform distribution over the interval \(a \le x \le b\) if its pdf is given by: \[f(x) = \frac{1}{b-a} \qquad \text{if } x \in [a,b]\] The cdf of \(X\) is: \[F(x) = \begin{cases}0 \quad &\text{if } x < a \\\frac{x-a}{b-a}\quad &\text{if } a \le x \le b \\ 1 &\text{if } x > b\end{cases}\] and the sdf of \(X\) is: \[S(x) = \begin{cases}1 \quad &\text{if } x < a \\\frac{b-x}{b-a}\quad &\text{if } a \le x \le b \\ 0 &\text{if } x > b\end{cases}\] If \(a = 0\) and \(b = 1\) then \(X\) is called the standard uniform random variable.

Properties: * \(\text E[X] = \frac{a+b}{2}\) * \(\text{Var}(X) = \frac{(b-a)^2}{12}\) * \((X|c < X < d) \sim U[c,d]\) * \((X-c|X > c) \sim U[0,b-c]\)

Example: Let \(X\) be the total time to process a passport application by the state department. It is known that X is uniformly distributed between 3 and 7 weeks. What is the probability that an application will be processed in 5 weeks or less?

Solution: The desired probability is: \[P\{X<5\} = \int_3^5 \frac{1}{4}dx = \frac{1}{2}\]