Question

Find Mean and Variance, if they exist in the following distribution:

fx={6x1-x   ,0<x<1 0  ,              elsewhere 

Answer 1

$$f(x) = \begin{cases} 6x(1-x) & 0<x<1 \\ 0 & elsewhere \end{cases}$$

$$E(X)=\displaystyle\int_{-\infin}^{\infin}xf(x)dx=\displaystyle\int_{0}^{1}6x^2(1-x)dx$$

$$=[2x^3-1.5x^4]\begin{matrix} 1 \\ 0 \end{matrix}=0.5$$

$$E(X^2)=\displaystyle\int_{-\infin}^{\infin}x^2f(x)dx=\displaystyle\int_{0}^{1}6x^3(1-x)dx$$

$$=[1.5x^4-1.2x^5]\begin{matrix} 1 \\ 0 \end{matrix}=0.3$$

$$Var(X)=E(X^2)-(E(X))^2=0.3-(0.5)^2=0.05$$