13 3 points The random variables X1 X2 and X3 are independen

13. 3 points The random variables X1, X2, and X3 are independent and are distributed as X^2(5), X^2(6), X^2(4), respectively. (a) 1 point Find E ( Xi), where pi is the product. (b) 2 points Find Var ( Xi).

Solution

Sol)

let x1, x2 and x3 are independent and are distributed as chisqaure(5),chisqaure(6),chisqaure(7),

ie., x1=chisqaure(5)

x2=chisqaure(6),

x3=chisqaure(7),

If X has the chi-square distribution with n degrees of freedom then

Mean=E(X)=n

var(X)=2n

A) E(X₁X₂X₃)

=E(X1)*E(X2)*E(X3)

where E(X1)=E(chisquare(5)) =5

E(X2)=E(chisquare(6)) =6

.E(X3)=E(chisquare(7)) =7

=5*6*4

=120

E(X₁X₂X₃)  =120

b) V(X₁X₂X₃)

₌V(X1)*V(X2)*V(X3)

where V(x1) =V(chisquare(5))= 2*5 =10

V(x2) =V(chisquare(6))= 2*6 =12

V(x3) =V(chisquare(7))= 2*7 =14

V(X₁X₂X₃) =10*12*14 =1680

V(X