Question

Find the variance of t? t 0 1 2 3 4 f(t) 1/9 2/9 3/9 2/9 1/9

Answer 1

$$\begin{matrix} t & & 0 & 1 & 2 & 3 & 4 \\ f(t) & & 1/9 & 2/9 & 3/9 & 2/9 & 1/9 \end{matrix}$$

$$E(T)=\mu=\dfrac{1}{9}(0)+\dfrac{2}{9}(1)+\dfrac{3}{9}(2)+\dfrac{2}{9}(3)+\dfrac{1}{9}(4)$$

$$=2$$

$$E(T^2)=\dfrac{1}{9}(0)^2+\dfrac{2}{9}(1)^2+\dfrac{3}{9}(2)^2+\dfrac{2}{9}(3)^2+\dfrac{1}{9}(4)^2$$

$$=\dfrac{16}{3}$$

$$Var(T)=E(T^2)-(E(T))^2=\dfrac{16}{3}-(2)^2=\dfrac{4}{3}$$

$$Var(T)=E((T-\mu)^2)$$

$$=\dfrac{1}{9}(0-2)^2+\dfrac{2}{9}(1-2)^2+\dfrac{3}{9}(2-2)^2+\dfrac{2}{9}(3-2)^2$$

$$+\dfrac{1}{9}(4-2)^2=\dfrac{4}{3}$$

$Var(T)=\dfrac{4}{3}$