Answer to Question #218574 in Statistics and Probability for Marcus Maraka

Question #218574
2 x 4
1
Expert's answer
2021-07-19T05:42:48-0400
f(x)={kx2x40otherwisef(x)=\begin{cases} kx & 2\leq x\leq 4 \\ 0 & otherwise \end{cases}

f(x)dx=24kxdx=[kx22]42\displaystyle\int_{-\infin}^{\infin}f(x)dx=\displaystyle\int_{2}^{4}kxdx=[\dfrac{kx^2}{2}]\begin{matrix} 4\\ 2 \end{matrix}


=16k24k2=6k=1=>k=16=\dfrac{16k}{2}-\dfrac{4k}{2}=6k=1=>k=\dfrac{1}{6}

E[X]=24x(16x)dx=[x318]42E[X]=\displaystyle\int_{2}^{4}x(\dfrac{1}{6}x)dx=[\dfrac{x^3}{18}]\begin{matrix} 4\\ 2 \end{matrix}


=6418818=289=\dfrac{64}{18}-\dfrac{8}{18}=\dfrac{28}{9}

E[X2]=24x2(16x)dx=[x424]42E[X^2]=\displaystyle\int_{2}^{4}x^2(\dfrac{1}{6}x)dx=[\dfrac{x^4}{24}]\begin{matrix} 4\\ 2 \end{matrix}


=256241624=10=\dfrac{256}{24}-\dfrac{16}{24}=10

Var(X)=σ2=E[X2](E[X])2=10(289)2=2681Var(X)=\sigma^2=E[X^2]-(E[X])^2=10-(\dfrac{28}{9})^2=\dfrac{26}{81}

σ=σ2=2690.5666\sigma=\sqrt{\sigma^2}=\dfrac{\sqrt{26}}{9}\approx0.5666


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