Homework Soursequestion about probability expectation conditioning Let Y1
question about probability / expectation / conditioning
Let Y1, Y2, ... be a sequence of iid random variables with all moments finite. Let N be a nonnegative-integer-valued random variable, independent from all the Yi, also with all moments finite. Let S = Y1 +... + YN, and define S = 0 if N = 0. Find the mean E[S] and the variance Var(S) in terms of the moments (mean, variance, second moment) of Y and N. Hint: begin by conditioning on the value of N. i.e. writeSolution
S=Y1 + Y2 +.........................+Yn
then
E(S)=E(Y1 +Y2 +......................+Yn )
= E(NYi ) as Yi \' s are iid
=E(N)*E(Yi ) since Yi and N are independent to each other
=m(N)1 * m(Y)1 where m(N)1 is first moment of N
m(S)1 is first moment of Yi \'s
E(S2 )=E((Y1 + Y2 +.....................+Yn )2 )
=E(NYi2 )+2E(NYi2) since Yi \'s are independent
=3m(N)1 * m(Y)2 where m(N)1 is first moment of N
m(Y)2 is second moment of Yi \'s
so var (S)=E(S2 ) - [E(S)]2
=m(N)1 *m(Y)2 - [n=m(N)1 *m(Y)2 ]2
