Question

Suppose f(x) = 1. 5 𝑥 for -1 < x <1. Determine the mean and variance of X

Answer 1

Check

$$\displaystyle\int_{-\infin}^{\infin}f(x)dx=\displaystyle\int_{-1}^{1}1.5x^2dx$$

$$=[0.5x^3]\begin{matrix} 1 \\ -1 \end{matrix}=1, True$$

$$\mu=E(X)=\displaystyle\int_{-\infin}^{\infin}xf(x)dx=\displaystyle\int_{-1}^{1}1.5x^3dx$$

$$=[\dfrac{1.5}{4}x^4]\begin{matrix} 1 \\ -1 \end{matrix}=0$$

$$E(X^2)=\displaystyle\int_{-\infin}^{\infin}x^2f(x)dx=\displaystyle\int_{-1}^{1}1.5x^4dx$$

$$=[0.3x^5]\begin{matrix} 1 \\ -1 \end{matrix}=0.6$$

$$Var(X)=E(X^2)-(E(X))^2$$

$$=0.6-(0)^2=0.6$$