hw5 - Homework Set No. 5 Probability Theory (235A), Fall...

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 2011 Due: Tuesday 11/01/11 at discussion section 1. (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 . 2. 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, E ( X- t ) 2 (the moment of inertia around t , which by the Huygens-Steiner theorem is simply a parabola in t , taking its minimum value of V...
View Full Document

Page1 / 3

hw5 - Homework Set No. 5 Probability Theory (235A), Fall...

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