7.4 The Uniform Distribution

For purposes of computing probabilities with ease, the uniform distribution is handy. Suppose that $X$ is a continuous random variable with p.d.f. $f(x).$ We say $X$ has a uniform distribution if there are real numbers $a$ and $b,$ with such that

We say that $X$ is uniformly distributed between $a$ and $b\mathit{,}$ and our shorthand notation for this distribution will be $X\sim U(a,b).$

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  In graphing the uniform distribution, over the interval $[a,b],$ draw a rectangle of height $1/(b-a)$ as shown in Figure 7.7.

  

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  If an $X$ is randomly generated, it will be between $a$ and $b,$ i.e., it won’t be less than $a$ or greater than $b.$

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  Since the total area must be 1, the height of the box is $1/(b-a).$

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  By symmetry, the mean $\mu$ of $X\sim U(a,b)$ is the midpoint of the interval $[a,b],$ i.e., it is the average of the two endpoints. In symbols,

  $$\mu =\frac{a+b}{2}.$$

  Knowing this is handy in interpreting some simulations.²⁹²⁹It turns out that if $X\sim U(a,b),$ then the variance of $X$ is ${\sigma }^2={(b-a)}^2/12.$ This too can be handy for some simulations.

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  Computing probabilities amounts to computing areas of rectangular boxes, as illustrated in the next example.