Let X1 X2 X3 be a sequence of independent and identically di

Let X1, X2, X3,... be a sequence of independent and identically distributed random variable with finite expectation E[X] and finite variance Var(X). Define

show that the sequence of random variables Y1,Y2,Y3,... converges in probability and express the limit in terms of E[X] and Var(X)

Solution

E(X) = p * sum (from 1->n) Xi
E(X^2) = p * sum (from 1->n) Xi^2

Var(X) = E(X^2)- (E(X))^2 = p * sum (from 1->n) Xi^2 - (E(X))^2
=>sum (from 1->n) Xi^2 = (Var(X) + (E(X))^2 )/p

Yn = 1/n * sum (from 1->n) Xi^2
= 1/n * (Var(X) + (E(X))^2 )/p
But p =1/n {Reason : X1, X2, X3,... be a sequence of independent and identically distributed random variable}
Therefore,
Yn = 1/n * (Var(X) + (E(X))^2 ) * n
= Var(X) + (E(X))^2 -----> Finite (Given : X1, X2, X3,... be a sequence of finite expectation E[X] and finite variance Var(X)).

Hence,
Yn = Var(X) + (E(X))^2 Converges and is Finite.