stat210a_fall07_hw1 - UC Berkeley Department of Statistics...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1 Fall 2007 Issued: Thursday, August 30 Due: Thursday, September 6 Notation: The symbols { d , p , m.s. } denote convergence in distribution, probability and mean-square respectively. Problem 1.1 Suppose X i is a sequence of independent random variables with a common mean E ( X i ) = for all i and different variances E ( X i ) 2 = 2 i . (a) Show that X n = P n i =1 X i n m.s. as long as n i =1 2 i = o ( n 2 ). Under the same assump- tions, does X n p ? Why or why not? (b) Now consider the estimate tildewide X n = P n i =1 1 i X i P n i =1 1 i . Prove that var( tildewide X n ) var( X n ). Use this to conclude that, under the same conditions as in part (a), tildewide X n p . Problem 1.2 Consider an i.i.d. sample { X 1 ,...,X n } from the uniform distribution on [0 , ], and the estimator M n = max { X 1 ,...,X n } ....
View Full Document

Page1 / 2

stat210a_fall07_hw1 - UC Berkeley Department of Statistics...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online