Question

Let X be a random variable with probability distribution

X -1 0 1 2 3

F(X) 0.125 .50 0.20 0.05 0.125

a) Find E(X) and VAR(X).

b) Find the probability distribution of the random variable Y= 2X+1. Using the 

probability distribution of Y determine E(Y) and VAR(Y).

Answer 1

a)

$$E(X)=0.125(-1)+0.50(0)+0.20(1)$$

$$+0.05(2)+0.125(3)=0.55$$

$$E(X^2)=0.125(-1)^2+0.50(0)^2+0.20(1)^2$$

$$+0.05(2)^2+0.125(3)^2=1.65$$

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

$$=1.65-0.55^2=1.3475$$

b)

$$\def\arraystretch{1.5} \begin{array}{c:c} y & -1 & 1 & 3 & 5 & 7 \\ \hline F(Y) & 0.125 & 0.50 & 0.20 & 0.05 & 0.125 \\ \end{array}$$

$$E(Y)=0.125(-1)+0.50(1)+0.20(3)$$

$$+0.05(5)+0.125(7)=2.1$$

$$E(Y)=2E(X)+1$$

$$E(Y^2)=0.125(-1)^2+0.50(1)^2+0.20(3)^2$$

$$+0.05(5)^2+0.125(7)^2=9.8$$

$$Var(Y)=E(Y^2)-(E(Y))^2$$

$$=9.8-2.1^2=5.39$$

$$Var(Y)=(2)^2Var(X)$$