** Class Activity **

1. Sketch the distribution $p_{X}(x)$

File:Example.jpg

2. What is the distribution's mean and standard deviation?
Mean:

${\text{Mean}}=E(x)$

$E(x)=\int _{0}^{2}xp(x)dx=\int _{0}^{2}x\left(1-{\frac {x}{2}}\right)dx={\frac {x^{2}}{2}}-{\frac {x^{3}}{6}}{\Big |}_{0}^{2}={\frac {2}{3}}$

Standard Deviation:

${\text{Standard Deviation}}={\sqrt {\text{Variance}}}$

${\text{Variance}}=\int _{0}^{2}x^{2}p(x)dx-[E(x)]^{2}=\left({\frac {x^{3}}{3}}-{\frac {x^{4}}{8}}\right){\Big |}_{0}^{2}-[E(x)]^{2}={\frac {6}{9}}-{\frac {4}{9}}={\frac {2}{9}}$

${\text{Standard Deviation}}={\sqrt {\text{Variance}}}={\sqrt {\frac {2}{9}}}$

3. What is its cumulative distribution function (CDF)?

File:Example.jpg