Lecture 4. E) Uniform

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Uniform Distribution on [math]\left[a,b\right][/math]

A r.v. [math]X[/math] follows a uniform distribution [math]U\left(a,b\right)[/math] if [math]X[/math] is continuous with pdf

[math]f_{X}\left(X\right)=\begin{cases} \frac{1}{b-a}, & x\in\left[a,b\right]\\ 0, & otherwise \end{cases}[/math]

Under the Uniform distribution, all values in [math]\left[a,b\right][/math] are “equally likely.”

Notice that if [math]X\sim U\left(a,b\right)[/math], then [math]X=\left(b-a\right)\widetilde{X}+a[/math] where [math]\widetilde{X}\sim U\left(0,1\right)[/math], and [math]f_{\widetilde{X}}\left(x\right)=1\left(x\in\left[0,1\right]\right)[/math].

Mean

[math]E\left(\widetilde{X}\right)=\int_{0}^{1}xdx=\frac{1}{2}[/math].

So, [math]E\left(X\right)=E\left(\left(b-a\right)\widetilde{X}+a\right)=\left(b-a\right)E\left(\widetilde{X}\right)+a=\frac{a+b}{2}[/math]

Variance

[math]Var\left(\widetilde{X}\right)=E\left(\widetilde{X}^{2}\right)-E\left(\widetilde{X}\right)^{2}=\int_{0}^{1}x^{2}dx-\left(\frac{1}{2}\right)^{2}=\frac{1}{3}-\frac{1}{4}=\frac{1}{12}[/math].

So, [math]Var\left(X\right)=Var\left(\left(b-a\right)\widetilde{X}+a\right)=\left(b-a\right)^{2}Var\left(\widetilde{X}\right)=\frac{\left(b-a\right)^{2}}{12}[/math].

MGF

[math]M_{X}\left(t\right)=\exp\left(at\right)M_{\widetilde{X}}\left(\left(b-a\right)t\right)=...=\frac{\exp\left(bt\right)-\exp\left(at\right)}{\left(b-a\right)t}[/math]