Question

A continuous random variable x has the pdf given by

f(x)={k(1-x) for 0<x<1 both inclusive

{o,elsewhere

find t,the value of the constant k. Also find the mean and the variance of x

Answer 1

$$\displaystyle\int_{-\infin}^{\infin}f(x)x=\displaystyle\int_{0}^{1}k(1-x)dx$$

$$=k[x-\dfrac{x^2}{2}]\begin{matrix} 1 \\ 0 \end{matrix}=\dfrac{1}{2}k=1$$

$$k=2$$

$$mean=\mu=E[X]=\displaystyle\int_{-\infin}^{\infin}xf(x)dx$$

$$=\displaystyle\int_{0}^{1}2x(1-x)dx=[x-\dfrac{2x^3}{3}]\begin{matrix} 1\\ 0 \end{matrix}=\dfrac{1}{3}$$

$$E[X^2]=\displaystyle\int_{-\infin}^{\infin}x^2f(x)dx=\displaystyle\int_{0}^{1}2x^2(1-x)dx$$

$$=[\dfrac{2x^3}{3}-\dfrac{x^4}{2}]\begin{matrix} 1 \\ 0 \end{matrix}=\dfrac{1}{6}$$

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

$$=\dfrac{1}{6}-(\dfrac{1}{3})^2=\dfrac{1}{18}$$