Class Activity
1. Sketch the distribution p X ( x ) {\displaystyle p_{X}(x)}
File:Example.jpg
2. What is the distribution's mean and standard deviation? Mean:
Mean = E ( x ) {\displaystyle {\text{Mean}}=E(x)}
E ( x ) = ∫ 0 2 x p ( x ) d x = ∫ 0 2 x ( 1 − x 2 ) d x = x 2 2 − x 3 6 | 0 2 = 2 3 {\displaystyle 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:
Standard Deviation = Variance {\displaystyle {\text{Standard Deviation}}={\sqrt {\text{Variance}}}}
Variance = ∫ 0 2 x 2 p ( x ) d x − [ E ( x ) ] 2 = ( x 3 3 − x 4 8 ) | 0 2 − [ E ( x ) ] 2 = 6 9 − 4 9 = 2 9 {\displaystyle {\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}}}
Standard Deviation = Variance = 2 9 {\displaystyle {\text{Standard Deviation}}={\sqrt {\text{Variance}}}={\sqrt {\frac {2}{9}}}}
3. What is its cumulative distribution function (CDF)?
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/615db7406d3a357de20a623732471d67a065e561)
