5 Let X1 X2 and X3 be independent random variables Suppose t

5. Let X1, X2, and X3 be independent random variables. Suppose the mean of X1 and X2 is 2, the mean of X3 is 5, the variance of X1 is 1, the variance of X2 is 2, and the variance of X3 is 5. (a) What is the mean and variance of X1 + X2 ? (b) What is the mean and variance of X1 2X2 + 3X3 ? (c) Define a new random variable X = 1 3 (X1 + X2 + X3). What is the mean and variance of X ? 1

Solution

Given

E(X₁)=2          Var(X₁)=1

E(X₂)=2         Var(X₂)=2

E(X₃)=5          Var(X₃)=5

(a)  

Mean and Varaince of X₁+X₂

    E(X₁+X₂)=E(X₁)+E(X₂)

                 =2+2

                 =4

     E(X₁+X₂)=4

Var(X₁+X₂)=V(X₁)+V(X₂)

                =1+2

               =3

Var(X₁+X₂)=3

(b)

Mean and variance of X₁-2X₂+3X₃

E(X₁-2X₂+3X₃)=E(X₁)-2E(X₂)+3E(X₃)

                     =2-2(2)+3(5)

                     =2-4+15

                    =13

E(X₁-2X₂+3X₃)=13

V(X₁-2X₂+3X₃)=V(X₁)-2V(X₂)+3V(X₃)

                     =1-2(2)+3(5)

                    =1-4+15

                   =12

V(X₁-2X₂+3X₃)=12

(c)

Mean and varaince of X=1 3 (X₁ + X₂ + _X3)

E(X)=13E(X1+X₂+X₃)

      =13{E(X1)+E(X2)+E(X3)}

      =13{2+2+5}

      =13{9}

      =117

E(X)=117

V(X)=1 3 V(X₁ + X₂ + X₃)

     =13{V(X1)+V(X2)+V(X3)}

      =13{1+2+5}

    =13{8}

      =104

V(X)=104