Assignment 1 consists of the following problems. The first three are extra probl
ID: 3004126 • Letter: A
Question
Assignment 1 consists of the following problems. The first three are extra problems not from the textbook. In these three problems, X_n represents a sequence of random variables and c is a fixed constant. Show that X_n rightarrow p c X_n rightarrow d c. If E(X_n) rightarrow c and Var(X_n) rightarrow 0 as n rightarrow infinity, then X_n rightarrow r = 2 c. (The symbol rightarrow r = 2 means convergence in quadratic means.) Use the result in (2) to show that if Y_1, Y_2,..., Y_n are iid copies of a random variable Y with E(Y) = mu and Var(Y) = sigma^2, let Y_n be the sample mean where Y_n = n^-1 summation i = 1 n Y_i, then Y_n rightarrow p mu (WLLN).Explanation / Answer
we are give that Xn is a sequence of random variables
and E(Xn) tends to c , where c is a constant
and when n --> infiinity , that is as n tends to infinity
Var(Xn) tends to 0 , that is at n --> infinity the sequence of random variables Xn tends to become 0.
and this is the case when Xn is quadratic in nature.
and Xn --> c { for r = 2 }
and the sequence Xn converges to 'c' , and c is a quadratic mean.