Question

Find the mean, variance, and standard deviation of the probability distribution of the random variable X, which

can take only the values 1,2 and 3, given that P(1)= 10/33, P(2)= 1/3, and P(3)= 12/33.

Answer 1

$$\begin{matrix} x & 1 & 2 & 3 \\ p(x) & 10/33 & 1/3 & 12/33 \end{matrix}$$

$$E(X)=1(\dfrac{10}{33})+2(\dfrac{1}{3})+3(\dfrac{12}{33})=\dfrac{68}{33}\approx2.0606$$

$$E(X^2)=1^2(\dfrac{10}{33})+2^2(\dfrac{1}{3})+3^2(\dfrac{12}{33})=\dfrac{162}{33}$$

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

$$=\dfrac{162}{33}-(\dfrac{68}{33})^2=\dfrac{722}{1089}$$

$$\sigma=\sqrt{\dfrac{722}{1089}}=\dfrac{\sqrt{722}}{33}\approx0.8142$$