Question

If a die is rolled six times, let X be then number the die obtained on the first roll and Y be the sum of the numbers obtained from all the rolls. Find the expected value and variance of x and y.

Answer 1

a)since each outcome has equal probability on dice :

P(X=i) =1/6 where i=1,2,3,4,5,6

mean E(X)=xP(x)= (1/6)*(1+2+3+4+5+6)=21/6 =7/2 =3.5

E(X²)=x²P(x)= (1/6)*(1^2+2^2+3^2+4^2+5^2+6^2)=91/6

therefore Variance (X) =E(X²)-(E(X))² =91/6-(7/2)² =35/12 =2.9167

b)

here Y=X1+X2+X3+X4+X5+X6

therefore mean of Y =E(Y)=E(X1)+E(X2)+E(X3)+E(X4)+E(X5)+E(X6)=3.5*6=21

since events are independent from die to die:

Var(Y) =Var(X1)+Var(X2)+Var(X3)+Var(X4)+Var(X5)+Var(X6)=(35/12)*6 =17.5