Question

X 3 6 9 12 15

P(X) 4/9 2/9 1/9 1/9 1/9

Answer 1

$$mean=E(X)=\sum_ix_ip(x_i)$$

$$=3(\dfrac{4}{9})+6(\dfrac{2}{9})+9(\dfrac{1}{9})+12(\dfrac{1}{9})+15(\dfrac{1}{9})$$

$$=\dfrac{20}{3}$$

$$E(X^2)=3^2(\dfrac{4}{9})+6^2(\dfrac{2}{9})+9^2(\dfrac{1}{9})+12^2(\dfrac{1}{9})$$

$$+15^2(\dfrac{1}{9})=62$$

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

$$=62-(\dfrac{20}{3})^2=\dfrac{158}{9}$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{\dfrac{158}{9}}=\dfrac{\sqrt{158}}{3}\approx4.19$$