ECE 541 Probability and Stochastic Processes; Fall 2008
Assignment 7; Due date: Thursday, Nov. 6, 2008
Text problems: Chapter 3: 22–24, 28; Chapter 4: 2, 6, 23, 26, 30(a); Chapter 5: 11, 14, 17;
Chapter 7: 26, 27, 34
Problem 1. For 1 ≤p < ∞, prove that convergence in Lpimplies convergence in probability.
Problem 2. Let X1, X2. . . , be an iid sequence, E[X1] = θ < ∞, and E[X2
1]<∞. For n≥1, let
ˆ
θn=n−1Pn
i=1 Xi. Calculate the mean and variance of ˆ
θn.
Problem 3. Continuing from the previous problem, show that ˆ
θnconverges to θin the mean-square
sense (or in L2). Provide a practical example to which this result can be applied to.
1