Answer to Question #228037 in Statistics and Probability for cat

Question #228037
A random variable X has the following density function fx(x) = 2kx, 0 < x < 1/k 

a) Find the constant k.

b) Find the mean.

c) Find the standard deviation.


1
Expert's answer
2021-08-23T15:30:50-0400

(a)


f(x)dx=01/k(2kx)dx=[2xkx22]1/k0\displaystyle\int_{-\infin}^{\infin}f(x)dx=\displaystyle\int_{0}^{1/k}(2-kx)dx=[2x-\dfrac{kx^2}{2}]\begin{matrix} 1/k \\ 0 \end{matrix}

=2/k1/(2k)=32k=1=>k=32=2/k-1/(2k)=\dfrac{3}{2k}=1=>k=\dfrac{3}{2}

(b)


E(X)=xf(x)dx=02/3(2x32x2)dxE(X)=\displaystyle\int_{-\infin}^{\infin}xf(x)dx=\displaystyle\int_{0}^{2/3}(2x-\dfrac{3}{2}x^2)dx

=[x2x32]2/30=827=[x^2-\dfrac{x^3}{2}]\begin{matrix} 2/3 \\ 0 \end{matrix}=\dfrac{8}{27}

(c)


E(X2)=x2f(x)dx=02/3(2x232x3)dxE(X^2)=\displaystyle\int_{-\infin}^{\infin}x^2f(x)dx=\displaystyle\int_{0}^{2/3}(2x^2-\dfrac{3}{2}x^3)dx

=[23x338x4]2/30=1081=[\dfrac{2}{3}x^3-\dfrac{3}{8}x^4]\begin{matrix} 2/3 \\ 0 \end{matrix}=\dfrac{10}{81}

Var(X)=σ2=E(X2)(E(X))2Var(X)=\sigma^2=E(X^2)-(E(X))^2

=1081(827)2=26729=\dfrac{10}{81}-(\dfrac{8}{27})^2=\dfrac{26}{729}

σ=σ2=2627\sigma=\sqrt{\sigma^2}=\dfrac{\sqrt{26}}{27}


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