Question

If F(x) 1/39(3x-2)2;0<_x_<3

O: elsewhere

1. Verify that F(x) is a PDF

2. find E(x) and Var(x)

Answer 1

1.

$$\displaystyle\int_{-\infin}^{\infin}F(x)dx=\displaystyle\int_{0}^{3}\dfrac{(3x-2)^2}{39}dx$$

$$=[\dfrac{(3x-2)^3}{3(3)(39)}]\begin{matrix} 3\\ 0 \end{matrix}=\dfrac{343+8}{351}=1, True$$

2.

$$E(X)=\displaystyle\int_{-\infin}^{\infin}F(x)xdx=\displaystyle\int_{0}^{3}\dfrac{x(3x-2)^2}{39}dx$$

$$=\displaystyle\int_{0}^{3}\dfrac{9x^3-12x^2+4x}{39}dx$$

$$=[\dfrac{1}{39}(\dfrac{9x^4}{4}-4x^3+2x^2)]\begin{matrix} 3\\ 0 \end{matrix}$$

$$=\dfrac{1}{39}(\dfrac{729}{4}-108+18-0)$$

$$=\dfrac{123}{52}$$

$$E(X^2)=\displaystyle\int_{-\infin}^{\infin}F(x)x^2dx=\displaystyle\int_{0}^{3}\dfrac{x^2(3x-2)^2}{39}dx$$

$$=\displaystyle\int_{0}^{3}\dfrac{9x^4-12x^3+4x^2}{39}dx$$

$$=[\dfrac{1}{39}(\dfrac{9x^5}{5}-3x^4+\dfrac{4x^3}{3})]\begin{matrix} 3\\ 0 \end{matrix}$$

$$=\dfrac{1}{39}(\dfrac{2187}{5}-243+36-0)$$

$$=\dfrac{384}{65}$$

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

$$=\dfrac{384}{65}-(\dfrac{123}{52})^2=\dfrac{4227}{13520}$$