Math 170B
Midterm 2
November 20
Answer the questions in the spaces provided on the question sheets. If you
run out of room for an answer, continue on the back of the page. Explain
your answers and reasoning.
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Midterm 1, Math 170B - Winter 2016
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When t
Useful formulas.
o PMF 0f Bernouli (p): pX(1) = p,pX(0) = p.
Expectation is p, variance is p(1 p).
o PMF 0f Bin(n,p):
Expectation is np. Variance is np(1 p).
o PMF of Geo(p):
1 1
Expectation is . Variance is 2 .
p p p
o PMF 0f P0i()\):
Ak
pX(k) : 6)H for
T. Liggett
Mathematics 170B Midterm 2 Solutions
May 23, 2012
(20) 1. (a) State Markovs inequality.
Solution: If X 0, then P (X a) EX/a for a > 0.
(b) Prove Markovs inequality.
Solution: a1cfw_Xa X. Taking expected values gives aP (X a) EX.
(c) Suppose X i
Math 170B - Probability Theory, Lec.1, Fall 2015 - Homework 5
Due: Wednesday, Oct. 28th 2015, at the beginning of class.
From the textbook, solve problems 4 and 5 at the end of Chapter 5. Moreover,
work through problem 2 (Chernov bound) which is solved in
Convergence of random variables
In this note, we explain the three different definitions of convergence of
random variables.
Let X1 , X2 , be a sequence of random variables which come from the
same experiment, that is, they are all functions from the same
Probability Theory, Math 170B, Winter 2016 - Homework 1
Extra problems:
Problem 1. Suppose that X1 and X2 are two independent exponential
random variables with parameters 1 and 2 respectively.
(a) What is the law of min(X1 , X2 )? (that is, find the PDF a
Math 170B - Probability Theory, Lec.1, Fall 2015 - Homework 7
Due: Friday(!), Nov. 13th 2015, at the beginning of class.
From the books supplementary problems, solve problems 18 (a), (c), (d), (e) and
19 in Chapter 7 (see http:/www.athenasc.com/prob-supp.
Identical distribution
In this note, we explain the concept of identical distribution.
Given two random variables X and Y (not necessary come from the same
experiment), that is,
X : 1 R
and
Y : 2 R,
where 1 might be different from 2 . But usually there is
Math 170B - Probability Theory, Lec.1, Fall 2015 - Homework 4
Due: Wednesday, Oct. 21st 2015, at the beginning of class.
From the textbook, solve problems 29, 30, 31, 32, 33 at the end of Chapter 4.
Moreover, solve the problems below:
Problem 1. Given a p
Useful formulas.
E[X|Y ] is the random variable which is equal to E[X|Y = k] when Y = k.
var(X|Y ) is the random variable which is equal to var[X|Y = k] when Y = k. Alternate
formula,
var(X|Y ) = E(X 2 |Y ) (E(X|Y )2 .
If Z is determined by values of Y th
Math 170B - Probability Theory, Lec.1, Fall 2015 - Homework 8
Due: Friday, Nov. 20th 2015, at the beginning of class.
From the textbook, solve problems 1, 2 and 3 at the end of Chapter 6.
From the books supplementary problems, solve problems 3, 6 (a), (b)
Probability Theory, Math 170B, Winter 2016 - Homework 5
Extra problems:
Problem 1. (a) Show that if
| maxcfw_0, a maxcfw_0, b| > 0
then
|a b| .
(b) Show that if Xn X in probability then maxcfw_0, Xn maxcfw_0, X in
probability.
Problem 2. Let x and y be t
Math 170B - Probability Theory, Lec.1, Fall 2015 - Homework 2
Due: Wednesday, Oct. 7th 2015, at the beginning of class.
From the books supplementary problems, solve problems 15, 16, 17, 30, 31, 32
in Chapter 4 (see http:/www.athenasc.com/prob-supp.html).
Math 170B Probability Theory: Lecture 13
Yuan Zhang
yuanzhang@math.ucla.edu
Department of Mathematics
UCLA
April 29th 2016
Yuan Zhang (Dept. Math, UCLA)
Probability Theory
April 29th 2016
1 / 11
Convergence in Probability
Here we will have another example
Math 170B Probability Theory: Lecture 20
Yuan Zhang
yuanzhang@math.ucla.edu
Department of Mathematics
UCLA
May 27th 2016
Yuan Zhang (Dept. Math, UCLA)
Probability Theory
May 27th 2016
1/8
Final Exam
Time: Wednesday June 8th, 2016, 11:30 AM - 2:30 PM
Locat
Math 170B Probability Theory: Lecture 11
Yuan Zhang
yuanzhang@math.ucla.edu
Department of Mathematics
UCLA
April 25th 2016
Yuan Zhang (Dept. Math, UCLA)
Probability Theory
April 25th 2016
1 / 11
Limit Theorems
In this chapter we will learn two very import
Math 170B Probability Theory: Lecture 12
Yuan Zhang
yuanzhang@math.ucla.edu
Department of Mathematics
UCLA
April 27th 2016
Yuan Zhang (Dept. Math, UCLA)
Probability Theory
April 27th 2016
1/9
Weak Law of Large Numbers
In the previous lecture, we used Mark
Math 170B Probability Theory: Lecture 7
Yuan Zhang
yuanzhang@math.ucla.edu
Department of Mathematics
UCLA
April 11th 2016
Yuan Zhang (Dept. Math, UCLA)
Probability Theory
April 11th 2016
1/9
Conditional Expectation
Finally, we have the following property
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