Practice exam for Midterm 1 (50 minutes)
SUBMITTED BY :
PUID#:
CLASS SECTION: 165, 166
Remarks:
(i) NO calculators, books or notes are allowed on this exam. Turn off all
electronic devices.
(ii) This exam contains 6 problems. The maximum possible score is
Name
Student ID #
Instructor:
Sergey Kirshner
STAT 416 Spring 2012
Practice Exam #2
March 20, 2012
You are not allowed to use books or notes. Non-programmable non-graphing calculators
are permitted. Please read the directions carefully. There are 8 proble
Version: January 10, 2011. Check web page for any updates.
Desire and Determination Bring Any Goal Within Reach Math 416 / Stat 416: Probability Spring 2011 Division 1 MWF 11:30am - 12:20pm (REC 121) Division 2 MWF 12:30pm - 1:20pm (UNIV 119)
Instructor:
Homework 8 (Due 04/10/2015)
Please show your detailed mathematical argument. Answers without
work will receive 0 point. You are allowed to submit your homework with a
partner. Staple your work if you have multiple pages, otherwise you will be
taken 2 poin
Practice exam for Midterm 1 (50 minutes)
SUBMITTED BY :
PUID#:
CLASS SECTION: 165, 166
Remarks:
(i) NO calculators, books or notes are allowed on this exam. Turn off all
electronic devices.
(ii) This exam contains 6 problems. The maximum possible score is
Homework 7 (Due 04/01/2015)
Please show your detailed mathematical argument. Answers without
work will receive 0 point. You are allowed to submit your homework with a
partner. Staple your work if you have multiple pages, otherwise you will be
taken 2 poin
Homework 5 Solutions
1. (4 points) Assume that the density function f of a continuous random
variable X has the form
x for x (0, 1)
f (x) =
0 otherwise,
where is some constant. Find the value of and the cumulative
distribution function of X.
R
R1
R1
Since
Solutions of HW 7
1. (a) I: One (natural) way to compute this probability is straightforward. Think of (U, V ) as a random point in the unit square
:= cfw_(x, y) R2 : x [0, 1], y [0, 1]. Since they are independent, their joint density
1 if (x, y)
fU,V (
Homework 8 (Due 04/10/2015)
Please show your detailed mathematical argument. Answers without
work will receive 0 point. You are allowed to submit your homework with a
partner. Staple your work if you have multiple pages, otherwise you will be
taken 2 poin
Solutions of Homework 6
1. (5 pts) Let X1 be uniform over (0, 1) and X2 is uniform over (0, 3).
Suppose they are independent. Find
(a) P (1 X1 + X2 2);
Since the two random variables are independent, the joint density
function is the product of the margin
Homework 1 (Due 01/28/2015)
1. Flip a fair coin three times. What is the probability that at least two
heads occurs? Describe the sample space and the event that we are
interested in, and solve the problem.
2. If three events A, B, C are independent, show
Solutions of HW4
1. Since A B, we have
P (AB)
P (A)
=
.
P (B)
P (B)
P (A|B) =
(1)
Let D1 , D2 denote the two outcomes. Then, B = cfw_D1 + D2 9 and
A = cfw_D1 + D2 = 10. Hence
P (A) =
=
=
6
X
i=1
6
X
i=4
6
X
i=4
P (B) =
6
X
P (D1 = i, D2 = 10 i)
P (D1 = i)
Remark: A typo in HW10: In Problem 1.9(c), the (2, 1)-entry should
be changed to 0.4. That is, the matrix should be
1
1 .6
2 .4
3 0
2
.4
.4
.2
3
0
.2 .
.8
Solution of HW9
All problems are from Pages 62-63 of Rick Durretts book Essentials of
Stochastic Pro
Solution of HW10
All problems are from Pages 62-63 of Rick Durretts book Essentials of
Stochastic Processes.
1.9(c) (Announced on Monday by email) Recall that there is a typo here: the
(2, 1)-entry should be changed to 0.4. That is, the matrix should read
MA416
Homework 3
Due Mar.14 2014
(Show your steps to get partial credits!)
Part I: chapter 6, 7 and 8
1. Determine whether the random variable is discrete or continuous.
(i) Time between oil changes on a car;
(ii) Number of heart beats per minute;
(iii) T
Examples for Chapter 5 Bayes theorem
1. Suppose an individual applying to a college determines that he has an 80% chance of
being accepted and he knows that dormitory housing will only be provided for 60% of
all the accepted students. Whats the probabilit
Examples for independent events
1. An oil exploration company currently has two active projects, one in Asia and the other
is Europe. Let A be the event that the Asian project is successful and B the event
that the European project is successful. Suppose
CHAPTER 4: CONDITIONAL PROBABILITY AND INDEPENDENCE
4.1 Conditional Probability and Trees
4.2 Independence of Events
4.1 Conditional Probability and Trees
Future is uncertain and, as new updates arrive, we must constantly reevaluate our
perception of
Homework 4 (Due 02/20/2015)
Please show your detailed mathematical argument. Answers without
work will receive 0 point. You are allowed to submit your homework with a
partner. Staple your work if you have multiple pages, otherwise you will be
taken 2 poin
8. Estimation theorems
Gameplan: Chapter 8.2 Markov inequality and Chebychev inequality Chapter 8.4
Jensens inequality and Chernov Bounds
Theorem 8.1 (Markov Inequality). Let X be a random variable such that P (X 0) = 1
(X only takes non-negative values)
7. The Central Limit Theorem
Section 6.3
Question 7.1. If X and Y are continuous random variables with densities f and g, what
is the density of Z = X + Y ?
1
Theorem 7.1. If X and Y are continuous random variables with densities f and g then
the density
9. Covariance and Correlation (7.7 in Ross)
Definition 9.1. The co-variance (covariance) between random variables X and Y is
Cov(X, Y ) = E[(X E[X])(Y E[Y ])]
= E[XY ] E[X]E[Y ]
Question 9.1. Assume that IA and IB are the indicator variables of two events
Math/Stat 416 Fall 2015 Instructor: Kelleher
7. Homework 7 Due 28 Oct. 2015
Do the following problems from the book.
Chapter 6 Problems - 6.7,6.19,6.21,6.27
Chapter 6 Theoretical Exercises - 6.9
Chapter 7 Problems - 7.6
As well as the following supplim