Final Exam (2 pages) Probability Theory (MATH 235A, Fall 2007) 1. (12 pts) One tosses a fair coin (H=Heads, T=Tails) until the rst appearance of the pattern HT. How many tosses on average is made? 2.
MATH 587
Assignment 2 Solutions
November 1, 2004
(1) If B B(R), then h1 (B ) = [h1 (B ) A] [h1 (B ) Ac ] = [f 1 (B ) A] [g 1 (B ) Ac ] F , so h is measurable. (2) Let B = cfw_B Rn |x + B B(Rn ). Sinc
MATH 587
Assignment 3
Due November 18, 2004
(1) Let be a measure, and 1 and 2 be signed measures on (, F ). Prove that (a) if 1 and 2 , then (1 + 2 ) . (b) 1 (c) if 1 (d) if 1 |1 | + , 1 1 . and 2 , t
MATH 587
Assignment 3 Solutions
November 18, 2004
(1) (a) Let (A) = (B ) = 0, |1 |(Ac ) = |2 |(B c ) = 0. Then (A B ) = 0 and |1 |(C ) = |2 |(C ) = 0, and so 1 (C ) = 2 (C ) = 0 for every C (A B )c .
MATH 587
Solutions to Assignment 4
December 2, 2003
1. (a) Consider a subsequence f (Xni , Yni , i 1. Since Xni X in P , there is a further subsequence Xnij , j 1 such that Xnij X a.s. Since Ynij Y in
MATH 587
Assignment 4
Due November 30, 2004
(1) Let X L2 (, F , P ) and let G be a -subalgebra of F . Show that among all G -measurable r.v.s Y L2 (, F , P ), there is a unique (a.s.) r.v. Y0 which mi
MATH 587
Solutions to Assignment 4
November 30, 2004
(1) E (Y X )2 = E [(Y E (X |G ) + (E (X |G ) X )]2 = E [Y E (X |G )]2 + E [(Y E (X |G )(E (X |G ) X )] + E [E (X |G ) X )]2 . But E [(Y E (X |G )(E
Quiz Probability Theory (MATH 235A, Fall 2007). October 5, 2007 1. Coin tossing A fair coin is tossed until for the rst time the same result appears twice in succession. Find the probability that the
Homework 4
Probability Theory (MATH 235A, Fall 2007)
1. Generating random variables with given distributions. Consider
a funciton F : R R that satises:
(i) F is nondecreasing and 0 F (x) 1 for all x;
Math493 - Fall 2017 - HW 1
Due 9/6/17
Renato Feres - Wash. U.
Preliminaries. Problems in probability theory can often be approached via stochastic simulation. A generic term
for problem-solving method
Math493 - Fall 2017 - HW 2
Renato Feres - Wash. U.
Preliminaries. In this assignment you will do a few more simulations in the style of the first assignment to explore
conditional probability, Bayes t
Math493 - Fall 2013 - HW 4
Renato Feres - Wash. U.
Preliminaries
We have up to this point ignored a central aspect of the Monte Carlo method: How to estimate errors? Clearly,
the larger the sample siz
Math493 - Fall 2017 - HW 3
Renato Feres - Wash. U.
Preliminaries
In this assignment we look a little bit further into the theory behind the Monte Carlo simulations we have been
using since the first h
Math493, Fall 2017 - HW 0
Renato Feres - Wash. U.
This homework assignment wont be collected, but do it as soon as possible. Its purpose is to have the R program
up and running and getting a first tas
Math493 - Fall 2017 - HW 5
Renato Feres - Wash. U.
Preliminaries
The purpose of this and next weeks homework assignment is to explore several methods for generating sample
values of random variables w
MATH 587
Assignment 2
Due October 26, 2004
(1) Let f and g be extended real-valued functions on (, F ), and dene h( ) = = f ( ) g ( ) if A, if Ac ,
where A is any set in F . Show that h is Borel measu
MATH 587
Assignment 2
Due October 26, 2004
(1) Let f and g be extended real-valued Borel measurable functions on (, F ), and dene h( ) = = f ( ) g ( ) if A, if Ac ,
where A is any set in F . Show that
Homework 3 Probability Theory (MATH 235A, Fall 2007) 1. Zero-one law. Let A1 , A2 , . . . be independent events. Prove that the events lim supn An and lim inf n An each have probabilities either zero
Homework 5 Probability Theory (MATH 235A, Fall 2007) 1. Random series. Let X1 , X2 , . . . be non-negative random variables. Prove the equality
E
n=1
Xn =
n=1
EXn .
Explain in what sense the equalit
Homework 6 Probability Theory (MATH 235A, Fall 2007) 1. Discrete independent random variables (a) Show that if X and Y are independent, integer valued random variables, then P(X + Y = n) =
k
P(X = k )
Midterm Exam Probability Theory (MATH 235A, Fall 2007) 1. (10 pts) Let X be a non-negative random variable, which takes values 0, 1, 2, . . . Prove that
EX =
n=1
P(X n).
2. (15 pts) From a set of n bo