Answer to Question #217700 in Statistics and Probability for mimay

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

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

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

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

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

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

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