Problem Set 4. Independence
1. Consider a standard 6-sided die and a biased die with sides (1,2,3,4,6,6), i.e. one which has no
5 and two 6s. Assume you pick one of the two dice, with equal probability, and roll it twice.
Define the events A1 = cfw_1st ro
STAB52 TUT 48
Quiz 1
7
September 14th,
2015
Please show all your work! Answers without supporting work will not be given credit.
Write answers in spaces provided. You have 50 minutes to complete this exam.
Name:
Student Number:
1. We have 5 balls, 3 red a
STAB52 Quiz 5
Please show all your work! Answers without supporting work will not be given credit. Write
answers in spaces provided. You have 50 minutes to complete this quiz.
Full Name:
Student Number:
Tutorial Number:
1. (5 pts) Let X and Y be independe
STAB52 Quiz 5
Please show all your work! Answers without supporting work will not be given credit. Write
answers in spaces provided. You have 50 minutes to complete this quiz.
Full Name:
Student Number:
Tutorial Number:
1. (5 pts) Let X and Y be independe
STAB52 - Introduction to Probability Quiz 4 (week 9)
Last Name:
Student #:
First Name:
Tutorial #:
Please show all your work! Answers without supporting work will not be
given credit. Write answers in spaces provided. You have 50 minutes to complete
this
STAB52 - Introduction to Probability Quiz 4 (week 9)
Last Name:
Student #:
First Name:
Tutorial #:
Please show all your work! Answers without supporting work will not be
given credit. Write answers in spaces provided. You have 50 minutes to complete
this
STAB52 Quiz 5
Please show all your work! Answers without supporting work will not be given credit. Write
answers in spaces provided. You have 50 minutes to complete this quiz.
Full Name:
Student Number:
Tutorial Number:
1. (5 pts) Let Z Z(0, 1) be a stand
STAB52 - Introduction to Probability Quiz 4 (week 9)
Last Name:
Student #:
First Name:
Tutorial #:
Please show all your work! Answers without supporting work will not be
given credit. Write answers in spaces provided. You have 50 minutes to complete
this
STAB52 Quiz 5
Please show all your work! Answers without supporting work will not be given credit. Write
answers in spaces provided. You have 50 minutes to complete this quiz.
Full Name:
Student Number:
Tutorial Number:
1. (5 pts) Let Z Z(0, 1) be a stand
STAB52 - Introduction to Probability Quiz 4 (week 9)
Last Name:
Student #:
First Name:
Tutorial #:
Please show all your work! Answers without supporting work will not be
given credit. Write answers in spaces provided. You have 50 minutes to complete
this
STAB52 WEEK 7 QUIZ
First Name: _
Student Number: _
Last Name: _
Score: _/20
Important: Please show all your work! Answers without supporting work will not be given credit.
Write answers in spaces provided. You have 50 minutes to complete this quiz.
1.
Let
First Name: _
Last Name: _
STAB52 WEEK 5 QUIZ
Student Number: _
Mark: _ /20
Important: Please show all your work! Answers without supporting work will not be given credit. Write
answers in spaces provided. You have 50 minutes to complete this exam
1. A fa
STAB52 Week 7 Quiz
Last Name: _
First Name: _
Score: _ / 20
Student Number: _
Please show all your work! Answers without supporting work will not be given credit. Write answers
in spaces provided. You have 50 minutes to complete this exam.
1. Suppose ther
STAB52 WEEK 5 QUIZ
Name:
Student #:
Score:
/20
Please show all your work! Answers without supporting work will not be given credit. Write
answers in spaces provided. You have 50 minutes to complete this exam.
Problem 1 [5 pts]: According to the CDC (Cente
STAB52 Week 7 quiz
Last Name: _
First Name: _
Student Number: _
You have 50 minutes to complete this exam. Please note that no aids are allowed.
1. For X ~ Geom(p), find E(2x) given that it is finite. Clearly state what value of p are for which
it is fini
STAB52 Quiz 2
Please show all your work! Answers without supporting work will not be given
credit. Write answers in spaces provided. You have 50 minutes to complete this
quiz.
Full Name:
Student Number:
Tutorial Number:
1. Let X Geometric(p), find P (X 2)
STAB52 - Introduction to Probability
Quiz 1
Last Name:
Student #:
First Name:
Tutorial #:
Rules: A non-programmable calculator is permitted.
Problem 1 (10 points)
Let A and B be events. The symmetric dierence A4B is dened to be the
set of all elements tha
STAB52H3 Week 7 Quiz
First Name: _ Student Number: _
Last Name: _ Mark: _ /20
Important: Please show all your work! Answers without supporting work will not be given credit. Write
answers in spaces provided. You have 50 minutes to complete this exam.
1. W
STAB52 Midterm Exam Solutions
Tuesday, Nov. 3, 2015
Problem 0. (i)
(20%) Let X be a random variable with finite expectation and variance.
Show that if a, b are real numbers, then V ar(aX + b) = a2 Var(X). Explain all steps.
Solution. Let = E[x]. By defini
Problem Set 6. Discrete Distributions
1. Suppose a symmetrical die is rolled 20 independent times, and each time we record whether or
not the event cfw_2,3,5,6 has occurred.
i What is the distribution of the number of times this event has occurred in the
Problem Set 3. Conditional Probability
Instructor: Prof. Sotirios Damouras
Teaching Assistant: Mohsen Soltanifar
September 17, 2016
1. Suppose we deal five cards from an ordinary 52-card deck.
i What is the conditional probability that all five cards are
Problem Set 5. Random Variables
1. Suppose that we roll two fair six-sided dice, and let Y be the sum of the two numbers showing.
Compute P (Y = y) for every real number y.
Solution: Define sample space S = cfw_(1, 1), (1, 2), (1, 3), . . . (6, 6) contain
Problem Set 7. Continuous Distributions
1. Let W Uniform(1,4). Compute each of the following:
i P (W 5)
ii P (W 2)
iii P (W 2 9) (Hint: If W 2 9, what must W be?)
iv P (W 2 2)
(
1/3
Solution: The PDF of the Uniform(1,4) is f (x) =
0
i P (W 5) =
R
R
0dx =
Problem Set 8. Important Distributions
1. Let X NegBinom(r, p) and Y NegBinom(s, p), where X and Y are defined on separate
and independent Bernoulli(p) trial sequences. If Z = X + Y , what is the distribution of Z?
Justify you answer.
2. (Hypergeometric d
Problem Set 4. Independence
1. Consider a standard 6-sided die and a biased die with sides (1,2,3,4,6,6), i.e. one which has no
5 and two 6s. Assume you pick one of the two dice, with equal probability, and roll it twice.
Define the events A1 = cfw_1st ro
Problem Set 6. Discrete Distributions
1. Suppose a symmetrical die is rolled 20 independent times, and each time we record whether or
not the event cfw_2,3,5,6 has occurred.
i What is the distribution of the number of times this event has occurred in the
Problem Set 5. Random Variables
1. Suppose that we roll two fair six-sided dice, and let Y be the sum of the two numbers showing.
Compute P (Y = y) for every real number y.
2. Suppose that a bowl contains 100 chips: 30 are labelled 1, 20 are labelled 2, a
Problem Set 7. Continuous Distributions
1. Let W Uniform(1,4). Compute each of the following:
i P (W 5)
ii P (W 2)
iii P (W 2 9) (Hint: If W 2 9, what must W be?)
iv P (W 2 2)
2. Find the values of the constant c for which the following functions are vali
STAB52H3 Midterm Test
SahITday,June 21, 2008
Duration: one hour and fifty minutes
(Sclwh"~
)
Please do NOT open this booklet until you are asked to do so
There are 9 pages including this page. Please check to see if you have all the pages.
Aids allowed: T
STAB52H Midterm Test
Saturday, June 9, 2007
Duration: one hour and fifty minutes
There are 8 pages including this page. Please check to see if you have all the pages.
Aids allowed: This test is open book. You are allowed to use a non-programmable
calculat
17. Sums and Order
Statistics
STAB52 F16
Sotirios Damouras
STAB52 F16
1
Sums & Order Statistics
Two specific transformations of RVs X1,X2 (X3,) are of practical
importance for Statistics & CS
Sums of RVs, i.e. X1+X2 (+X3+)
Model aggregate behavior, e.g
19. Moment Generating
Functions
STAB52 F16
Sotirios Damouras
STAB52 F16
1
Moment Generating Function
The Moment Generating Function (MGF) of RV X is given by
m(t ) E (e )
tX
Well-defined when m(t) is finite t < , for some >0
MGF provides alternative wa
19. Moment Generating
Functions
STAB52 F16
Sotirios Damouras
STAB52 F16
1
Moment Generating Function
The Moment Generating Function (MGF) of RV X is given by
m(t ) E (e )
tX
Well-defined when m(t) is finite t < , for some >0
MGF provides alternative wa
STAB52 F16 - LEC02
Quiz 7
Name (f,l):
ID #:
(
cex+y , x, y > 0, x + y < 1
1. Let the RVs X, Y have joint PDF fX,Y (x, y) =
0,
otherwise
i. Find the value of c.
ii. Find the marginal PDF of X.
Solution:
i.
ZZ
1
Z
Z
1x
x+y
ce
fX,Y (x, y)dxdy =
R2
0
dy dx
0
Problem Set 18. Moments
1. (Q3.2.4, p.147 in E&R)
(
6xy + 9x2 y 2 /2,
Let X, Y have joint PDF f (x, y) =
0,
following:
i.
ii.
iii.
iv.
v.
vi.
vii.
0yx1
Compute each of the
o/w
E(X)
E(Y )
E(X 2 )
E(Y 2 )
E(XY )
Cov(X, Y )
Corr(X, Y )
Solution:
ii.
iii.
iv.