Continuous Uniform Distribution

Topics: Probability - Distribution Function

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(definition)

Let $X$ be an absolutely continuous random variable such that $R_X=[a,b]$, where all intervals of a given length have the same probability.

In such a case, we say that $X$ has a continuous uniform distribution (or rectangular distribution) and write $X\sim \mathrm{Unif}([a,b])$.

Continuous Version

Notice that this is just the continuous version of the discrete uniform distribution.

Density Function §

(theorem)

The density function of such an $X$ is given by:

$$f_X(x)=\{\begin{array}{ll}\frac{1}{b-a}& \text{if }a\le x\le b\\ 0& \text{otherwise}\end{array}$$

Distribution Function §

(theorem)

Such an $X$’s distribution function is:

$$F_X(x)=\{\begin{array}{ll}0& \text{if }x<a\\ \frac{x-a}{b-a}& \text{if }a\le x<b\\ 1& \text{if }x\ge b\end{array}$$

Expected Value §

(theorem)

When it comes to the expected value of such an $X$, it is:

$$\mathbb{E}[X]=\frac{a+b}{2}$$

Variance §

(theorem)

As for the variance of such an $X$, it is:

$$\mathrm{Var}(X)=\frac{(b-a{)}^2}{12}$$

Moment-Generating Function §

(theorem)

The moment-generating function of such an $X$ is:

$$M_X(t)=\frac{e^{tb}-e^{ta}}{t(b-a)}$$