Continuous Uniform Distribution

Figure 1: The density function for a random variable on the interval $[1,3]$.

Example. Let $T$ be the waiting for a bus when a bus comes every 30 min,

$$f (t) = \frac{1}{30},~ 0 \leq t \leq 30$$

• Example 6.1: It is assumed that length $X$ of a conference has a uniform distribution on the interval $[0, 4]$.

  i

  What is the probability density function?

  $$f(x)=\left\lbrace \begin{array}{l} \frac{1}{4},~0 \leq x \leq 4\\ 0,~elsewhere\ \end{array}\right.$$

  ii

  What is the probability that any given conference lasts at least 3 hours?

  $$P(X\geq 3)=\int_3^4\frac{1}{4}dx=\frac{1}{4}$$

• Theorem 6.1:
  
• Mean is at the center of the range as we would expect.