PSTAT 120A: Discussion Session 2 (Apr 4 - Apr 8)
Practice problems:
Problem 0. Problems from Lectures 2-3.
Problem 1. A family has two children.
(a) What is the probability that both are boys, given that the younger child is a boy?
(b) What is the probabi

PSTAT 120A: Probability and Statistics
Winter 2015
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 functi

FINAL EXAM A
PSTAT 120A - Winter 2014
By writing my name and signing below I swear that I have not engaged in any form of
academic dishonesty or cheating.
Name:
Signature:
Section meeting time:
Do not turn the page until you are directed to do so.
Turn

MIDTERM EXAM A
PSTAT 120A - Winter 2014
By writing my name and signing below I swear that I have not engaged in any form of
academic dishonesty or cheating.
Name:
Signature:
Section meeting time:
Do not turn the page until you are directed to do so.
Tur

FINAL EXAM A
PSTAT 120A - Spring 2013
By writing my name and signing below I swear that I have not engaged in any form of
academic dishonesty or cheating.
Name:
Signature:
Section time and TAs name:
Do not turn the page until you are directed to do so.

Wenjian Liu (Instructor)
PSTAT 120A Winter 2015
Homework 7
Due Tuesday, Mar 10, 2015
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 ?
Problem 7.2
Screws produced

Wenjian Liu (Instructor)
PSTAT 120A Winter 2015
Homework 3
Due Tuesday, Feb 3, 2015
Problem 3.1
What is the probability of rolling an even number with a single die, given the die roll is 3 or less?
Problem 3.2
Toss a fair coin 10 times. If you know that a

Wenjian Liu (Instructor)
PSTAT 120A Winter 2015
Homework 2
Due Tuesday, Jan 20, 2015
Problem 2.1
Before the early 1990s, a telephone area code in the US consisted of three digits, where the first was
not 0 or 1, the second was 0 or 1, and the third was an

PSTAT 120A
Fall 2015
Final
Name (Print):
Discussion Section:
Teaching Assistant:
Do not turn the page until you are directed to do so.
Turn your cellphone off and put it away during the exam period.
You have 180 minutes to complete the exam.
You may u

PSTAT 120A: Week 7 - Solutions
Practice problems:
Problem 1. Let U U (0, 1) be the breakpoint and L the length of the shorter piece. Then L is
a function of the breakpoint U (L depends on U ),
(
U,
if U < 0.5,
L(U ) = 1 U, if U > 0.5.
and we can compute i

Lecture 13
1
Expected values
1.1
Expected values and joint distributions
Recall that for a continuous random variable X with PDF fX , the expected value of X is
Z
xfX (x)dx.
E(X) =
The expected value of the random variable g(X) (a function of X, for exam

Lecture 11
1
Joint distributions (Pitman 3.1 and 5.2)
So far we have considered the distribution of a single random variable in isolation, but in many
situations we need to understand the probabilistic relationship between several random variables.
For ex

PSTAT 120A: Week 6 (Oct 31 - Nov 4)
Practice problems:
Problem 1. Suppose X has probability density function (PDF)
( 2
cx (1 x)2 , if 0 < x < 1,
f (x) =
0,
otherwise.
where c is a constant. Find:
(a) c
(b) the cumulative distribution (CDF) function of X
(

PSTAT 120A: Week 1 (Sep 26 - Sep 30)
Practice problems:
Problem 1. I have three fair coins and flip each of them one time.
(a) Describe the sample space
(b) How many possible outcomes are there?
(c) What is the probability of getting at least two heads?
(

PSTAT 120A: Discussion Session 8 (Nov 14 - Nov 18)
Practice problems:
Problem 1. Select a point (X, Y ) at random (i.e. uniformly) from the unit circle with center (0, 0)
and radius 1, and let R be the distance of the point from the origin.
(a) Find the p

PSTAT 120A: Discussion Session 3 (Oct 10 - Oct 14)
Practice problems:
Problem 1. A box contains 50 red and 30 blue balls. Four balls are drawn at random from the
box, one after another. Find the probability that all four are red if we sample:
(a) Without

PSTAT 120A: Discussion Session 2 (Oct 3 - Oct 7)
Practice problems:
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) 0 = 1 and 1 = n
n
n
(b) k = n k
P
n
(c) nk=0 k = 2n
n
n
1

PSTAT 120A: Discussion Session 3 - Solutions
Practice problems:
Problem 1. Let X be the number of red balls drawn from the box.
(a) If the draws are done without replacement, then X HypGeo(N, n, m) with parameters N = 80,
m = 50, and n = 4 (see Lecture 6)

PSTAT 120A: Week 6 - Solutions
Practice problems:
Problem 1.
(a) The PDF should be nonnegative and integrate to 1, so from
Z 1
Z
c
cx2 (1 x)2 = ,
f (x)dx =
30
0
we obtain c = 30.
(b) The CDF is obtained by integrating the PDF:
if x < 0,
Z x
0,
5
4
3
6x

PSTAT 120A: Discussion Session 7 (Nov 7 - Nov 11)
Practice problems:
Problem 1. You have a stick of length 1.
(a) You break the stick in two parts. The point where you break it is uniformly distributed on the
stick. What is the expected length of the shor

PSTAT 120A
Fall 2016
Midterm
Name (Print):
Discussion Section:
Teaching Assistant:
Do not turn the page until you are directed to do so.
Turn your cellphone off and put it away during the exam period.
You have 75 minutes to complete the exam.
You may

PSTAT 120A: Week 1 - Solutions
Practice problems:
Problem 1.
(a) = cfw_(H, H, H), (H, H, T ), (H, T, H), (T, H, H), (T, T, H), (T, H, T ), (H, T, T ), (T, T, T )
(b) There are 2 2 2 = 8 possible outcomes.
(c) Let A =two heads and one tail = cfw_(H, H, T )

PSTAT 120A: Discussion Session 4 (Oct 17 - Oct 21)
Practice problems (there will not be a quiz in Week 5):
Problem 1. A small commuter plane has 30 seats. The probability that any particular passenger
will not show up for a flight is 0.10, independently o

PSTAT 120A: Discussion Session 2 - Solutions
Practice problems:
Problem 1.
(a) We can choose no element in one way and exactly one element in n ways.
(b) This identity states that it is equivalent to:
- choose k elements to include in a combination
- choo

PSTAT 120A
Homework 6
Spring 2016
Problem 1: Suppose that in a group of 10 people, each person has a 0.1 probability of
having a certain disease, independently from person to person. Now consider a testing plan
where the group is split into two groups of

PSTAT 120A
Instructor
Syllabus
Spring 2016
Wade Herndon
herndon@pstat.ucsb.edu
Office: South Hall 5519
Office Hours TR 11:00-1:45, or by appointment
Your TA will also hold office hours.
Course
Information
Lecture: 9:30-10:45 TR in IV Theater 2
Website: ga

PSTAT 120A
Homework 5
Winter 2016
Due: Thursday, 2/18/2016 by 4:00 pm
Problem 1: Prove that if a continuous random variable has an exponential PDF with
parameter , then for every pair of positive real numbers s and t, Pr (X > s + t | X > s) =
Pr (X > t).

PSTAT 120A
Homework 1
Winter 2016
Due: Thursday, 1/14/2016 by 4:00 pm
Problem 1: Give the sample space for each of the following experiments:
(a) An election decides between two candidates A and B.
= cfw_Candidate A wins, Candidate B wins.
(b) A two side

PSTAT 120A: Discussion Session 4 - Solutions
Practice problems:
Problem 1.
(a) The PDF should be nonnegative and integrate to 1, so from
Z 1
Z
c
cx2 (1 x)2 = ,
f (x)dx =
30
0
we obtain c = 30.
(b) The CDF is obtained by integrating the PDF:
if x < 0,
Z x