Hw5 - Homework Set No 5 – Probability Theory(235A Fall 2009 Posted — Due 1 Prove that if X is a random variable that is independent of itself

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: Homework Set No. 5 – Probability Theory (235A), Fall 2009 Posted: 10/27/09 — Due: 11/3/09 1. Prove that if X is a random variable that is independent of itself, then there is a constant c ∈ R such that P ( X = c ) = 1. 2. (a) If X ≥ 0 is a nonnegative r.v. with distribution function F , show that E ( X ) = Z ∞ P ( X ≥ x ) dx. (b) Prove that if X 1 ,X 2 ,..., is a sequence of independent and identically distributed (“i.i.d.”) r.v.’s, then P ( | X n | ≥ n i.o.) = 0 if E | X 1 | < ∞ , 1 if E | X 1 | = ∞ . (c) Deduce the following converse to the Strong Law of Large Numbers in the case of undefined expectations: If X 1 ,X 2 ,... are i.i.d. and E X 1 is undefined (meaning that E X 1+ = E X 1- = ∞ ) then P lim n →∞ 1 n n X k =1 X k does not exist ! = 1 . 3. Let X be a r.v. with finite variance, and define a function M ( t ) = E | X- t | , the “mean absolute deviation of X from t ”. The goal of this question is to show that the function M ( t ), like its easier to understand and better-behaved cousin,...
View Full Document

This note was uploaded on 11/13/2011 for the course STAT 235A taught by Professor Danromik during the Fall '11 term at UC Davis.

Page1 / 3

Hw5 - Homework Set No 5 – Probability Theory(235A Fall 2009 Posted — Due 1 Prove that if X is a random variable that is independent of itself

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