Wenjian Liu (Instructor)
PSTAT 120A Fall 2014
Solutions of Homework 6
Due Wednesday, Dec 3, 2014
Problem 6.1
Roll a fair die. Dene X = number of dots on the side that comes up. Calculate EX and E(X 2 ).
Proof.
6
E(X) =
iP (X = i)
i=1
6
=
i=1
i
6
1 6(6 + 1

Wenjian Liu (Instructor)
PSTAT 120A Fall 2014
Solutions of Homework 4Due Wednesday, Nov 12, 2014
Problem 4.1
Fischer and Spassky play a chess match in which the rst player to win a game wins the match. After
10 successive draws, the match is declared draw

PSTAT 296 A-B
Fall/Winter 2016-17
Research Opportunities for Top PSTAT Undergraduates for 2016-17
To provide research opportunities for our majors, PSTAT 296A-B, Research Projects in Actuarial
Science, is now open to top undergraduates in Actuarial Scienc

Wenjian Liu (Instructor)
PSTAT 120A Fall 2014
Solutions of Homework 3 Due Wednesday, Oct 29, 2014
Problem 3.1
What is the probability of rolling an even number with a single die, given the die roll is 3 or less?
Proof. Write A = cfw_the rolling number is

PSTAT 120A
Probability and Statistics
Fall 2014
Final Examination Practice 2
P#1
10
P#2
10
P#3
10
P#4
10
P#5
15
P#6
15
P#7
10
P#8
10
P#9
10
P#10
20
Total
100
Please note that the full credit of this nal is 100, although these 10 questions are
worth 120 po

Wenjian Liu (Instructor)
PSTAT 120A Fall 2014
Solutions of Homework 5
Due Monday, Nov 24, 2014
Problem 5.1
Let X and Y be continuous random variables with joint density function
2 y
if 0 x y,
e
f (x, y) =
0
otherwise.
(a) Find the marginal densities of

Wenjian Liu (Instructor)
PSTAT 120A Fall 2014
Solutions of Homework 8
Due Thursday, Dec 18, 2014
Problem 8.1
A uniform random number X divides [0, 1] into two segments. Let R be the ratio of the smaller versus
the larger segment. Compute the density of R.

Wenjian Liu (Instructor)
PSTAT 120A Fall 2014
Solutions of Homework 7 Due Wednesday, Dec 10, 2014
Problem 7.1
Let X1 and X2 be independent random variables that are uniformly distributed on cfw_1, . . . , n. What
is the PMF of S = X1 + X2 ?
Proof. First t

PSTAT 120A: Probability and Statistics
Fall 2014
Lecture 0: Summation
Lecturer: Wenjian Liu
0.1
Scribes: WL
Summation
In discrete probability models, it is inevitable to meet the summations, either nite or innite.
Here it is necessary to introduce some ba

PSTAT 120A
Probability and Statistics
Spring 2015
Midterm Examination Practice 1
P#1
20
P#2
20
P#3
20
P#4
20
P#5
20
P#6
30
Total 100
Please note that the full credit of this midterm is 100, although these 6 questions are worth 130 points in all. And you t

Lecture 5
Discrete Random Variables
Probability Mass Functions (PMF)
Cumulative Distribution Functions (CDF)
Expectation (mean) and Variance
Sveinn lafsson
olafsson@pstat.ucsb.edu
Random Variables
- A random variable X is a real-valued function whose doma

Lecture 6
The Bernoulli and Binomial distributions
Sveinn lafsson
olafsson@pstat.ucsb.edu
Bernoulli Distribution
Bernoulli random variables describe experiments that
either result in a success or a failure.
Examples:
Heads (success) or tails (failure) in

Lecture 7
Poisson Distribution, Poisson Approximation
to the Binomial distribution
Hypergeometric Distribution, Binomial
Approximation to the Hypergeometric
Sveinn lafsson
olafsson@pstat.ucsb.edu
Poisson Distribution
Sometimes we are interested in the num

PSTAT 120A: HW2
Due Oct 20 at the beginning of class
Clearly mark your solutions with your name, the name of your TA, and the time
of your discussion session.
Make sure to staple the pages of your solution set together.
Problem 1.
(a) A restaurant oers 15

PSTAT 120A: Discussion Session 3 (Oct 13 - Oct 15)
Problem 1. Explain (no need to derive) why the following identities are true for any n and k such
that 0 k n:
n
n
(a) k = n k .
(b)
n
k=0
n
k
= 2n [Hint: recall that the number of subsets of a set of n el

Lecture 3
(I) Independence, Bayes Formula, Tree Diagrams
(II) Basic Counting Principle
Sveinn lafsson
olafsson@pstat.ucsb.edu
Independence
Intuitively, we say that two events are independent if the
occurrence of one of the events gives us NO information

Lecture 4
Permutations and Combinations
Sveinn lafsson
olafsson@pstat.ucsb.edu
Permutations
We use Permutations when we are interested in the number of
possible ways to choose a subset of objects, and ORDER IS
IMPORTANT (Ex: a,b,b,a is not the same as b,

PSTAT 160B Winter 2017
Homework 5
Solve the exercises (1)-(9) below, and submit only the Python exercise (10) on Thursday February
23rd. The solution to (1)-(9) will be posted on Gauchospace.
Let us denote by B(t) , t 0 a standard Brownian motion on a pro

Wenjian Liu (Instructor)
PSTAT 120A Fall 2014
Solutions of Homework 1
Due Monday, Oct 13, 2014
Problem 1.1
Suppose that a number x is to be selected from the real line R, and let A, B, and C be the events
represented by the following subsets of R, where t

Wenjian Liu (Instructor)
PSTAT 120A Fall 2014
Solution of Homework 2
Due Monday, Oct 20, 2014
Problem 2.1
Before the early 1990s, a telephone area code in the US consisted of three digits, where the rst was
not 0 or 1, the second was 0 or 1, and the third

PSTAT 120A
Probability and Statistics
Fall 2014
Final Examination Practice 1
P#1
10
P#2
10
P#3
10
P#4
15
P#5
10
P#6
10
P#7
10
P#8
15
P#9
10
P#10
20
Total
100
Please note that the full credit of this nal is 100, although these 10 questions are
worth 120 po

ACTUARIAL ACCOUNT
FALL NEWSLETTER
Issue 3, Volume 1
UCSB Actuarial Association 2016
IN THIS ISSUE
Logistics:
Graduating Students (1)
Summary of Fall Quarter Events (1-3)
Interview with Donald Hsu- Aon Representative (3)
University of California,
Santa Bar

Math 394 B&C. Probability I. Summer 2014. Homework 7, due August 13
Homework 7 Solution, due August 13
Problem 3. Calculate EX and Var X for X v Exp().
Solution. Density:
(
ex , x > 0;
p(x) =
0, x 0.
Therefore,
Z
xex dx.
EX =
0
x
Integrate it by parts: le

PSTAT 120B, Spring 2016
HW6 Solutions
Updated: 05/09/2016
Homework 6 Solutions
Solution:
(i) We find the method of moments estimator by equating the theoretical moments to the sample
moments:
Z 1
Z 1
+ 1 set
E(Y ) =
yf (y)dy
( + 1)y +1 dy =
=Y
+2
0
0
2Y

PSTAT 120B, Spring 2016
HW8 Solutions
Updated: 05/28/2016
Homework 8 Solutions
Solution: This problem fits into a general scenario in which we consider confidence intervals of the
form z/2 and find n such that z/2 B where B is the desired bound (see Secti

PSTAT 120B, Spring 2016
HW5 Solutions
Updated: 05/04/2016
Homework 5 Solutions
Solution:
(a) Yi Uniform(, + 1) = E(Yi ) = + 12 and 2 = V ar(Yi ) = 1/12. Moreover, Yi s have common
CDF and PDF F (y) = y and f (y) = 1 for < y < + 1.
fY(n) (y) = n(F (y)n1 f

PSTAT 120B, Spring 2016
HW7 Solutions
Updated: 05/21/2016
Homework 7 Solutions
i.i.d.
i.i.d.
Solution: We suppose X1 , ., Xn N (1 , 2 ) and Y1 , ., Yn N (2 , 2 ) are two independent normal
samples with equal variance and we want to determine if,
Pn
2 Pn

PSTAT 120A: Probability and Statistics
Fall 2014
Lecture 1: Week 1
Lecturer: Wenjian Liu
1.1
Scribes: WL
Introduction to probability
1.1.1
Introduction
The theory of probability has always been associated with gambling and many most accessible
examples st

PSTAT 120A: Probability and Statistics
Fall 2014
Lecture 2: Week 2
Lecturer: Wenjian Liu
2.1
Scribes: WL
Combinations
First lets recall that a permutation is an ordered arrangement of r objects from a set of n objects.
Ex. How many ways can we choose a Pr

PSTAT 120A: Probability and Statistics
Fall 2014
Lecture 4: Week 4
Lecturer: Wenjian Liu
4.1
4.1.1
Scribes: WL
Random Variables
Introduction
In many situations we are not concerned directly with the outcome of an experiment, but instead
with some function

PSTAT 160B Spring 2016
Homework 6
(1). A job shop consists of three machines and two repairmen. The amount of time a machine
works before breaking down is exponentially distributed with mean 10. If the amount of time it
takes a single repairman to fix a m