Math 151 Probability
Spring 2011
Jo Hardin
Distributions
Random Variables & Distributions
random variable Let S be the sample space for an experiment. A real-valued function
that is dened on S is called a random variable.
distribution Let X be a random
Stats for Clinical Trials, Math 150
Jo Hardin
Homework # 5, Due: Thurs, 10/7/10
The following questions relate the to the sepsis.csv data. [Bernard, G. R., A. P.
Wheeler, et al. (1997). The eects of ibuprofen on the physiology and survival of patients
wit
Stats for Clinical Trials, Math 150
Jo Hardin
Homework # 2, Due: Thurs, 9/18/10
1. Eisenhofer et al. (1999) investigated the use of plasma normetanephrine and metanephrine
for detecting pheochromocytoma in patients with von Hippel-Lindau disease and multi
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #1
1. Let C denote a sample space and A be a subset of C. Establish the following
set theoretic identities:
(a) A = ,
(b) A = A;
where denotes the empty set. Justify your steps.
Solution:
(a) Proof: The
Math 151. ProbabilityRumbos
Spring 2014
Review Problems for Exam 2
(1) Let X and Y be independent Exponential(1) random variables. Put Z =
Y
. Compute
X
the distribution functions FZ (z) and fZ (z).
(2) A random point (X, Y ) is distributed uniformly on t
Math 151. Rumbos
Spring 2014
1
Review Problems for Exam 3
1. Suppose that a book with n pages contains on average misprints per page.
What is the probability that there will be at least m pages which contain more
than k missprints?
2. Suppose that the tot
Math 151. Rumbos
Spring 2014
1
Topics for Final Exam
1. Probability Spaces
1.1 Sample spaces
1.2 Fields
1.3 Probability functions
1.4 Independent events
1.5 Conditional probability
2. Random Variables
2.1 Continuous and discrete random variables
2.2 Cumul
Math 151. Rumbos
Spring 2014
1
Exam 3 (Part I)
Friday, May 2, 2014
Name:
This is the inclass portion of Exam 3. This is a closedbook and closednotes
exam; you may consult only the Special Distributions and the Normal Distribution
Probabilities Table hando
Math 151. Rumbos
Spring 2014
1
Exam 3 (Part II)
Due on Monday, May 5, 2014
Name:
This is the outofclass portion of Exam 3. There are three questions in this portion
of the exam. This is a closedbook and closednotes exam; you may consult only the
Special D
Math 151. ProbabilityRumbos
Spring 2014
Topics for Exam 2
1. Expectations of Random Variables
1.1. Expected Value a random variable
1.2. Expected value of functions of random variables
1.3. Moments and moment generating function
1.4. Variance of a random
Math 151. Rumbos
Spring 2014
1
Exam 2 (Part I)
Wednesday, March 26, 2014
Name:
This is the inclass portion of Exam 2. This is a closedbook and closednotes exam;
you may consult only the Special Distributions handout.
Show all signicant work and give reaso
Math 151. Rumbos
Spring 2014
1
Exam 2 (Part II)
Due on Monday, March 31, 2014
Name:
This is the outofclass portion of Exam 2. There are three questions in this portion
of the exam. This is a closedbook and closednotes exam; you may consult only the
Specia
Math 151. Rumbos
Spring 2014
1
Exam 1 (Part I)
Friday, February 21, 2014
Name:
This is the inclass portion of Exam 1. This is a closedbook and closednotes exam;
you may consult only the Special Distributions handout.
Show all signicant work and give reaso
Math 151. Rumbos
Spring 2014
1
Exam 1 (Part II)
This is the outofclass portion of Exam 1. This is a closedbook and closednotes
exam; you may consult only the Special Distributions handout. You may work on
these questions as long as you wish. Show all sign
Math 151.
ProbabilityRumbos
Spring 2014
Review Problems for Exam 1
(1) There are 5 red chips and 3 blue chips in a bowl. The red chips are numbered 1, 2, 3, 4, 5 respectively,
and the blue chips are numbered 1, 2, 3 respectively. If two chips are to be dr
Math 151. ProbabilityRumbos
Spring 2014
Topics for Exam 1
1. Probability Spaces
1.1. Sample spaces
1.2. Fields
1.3. Probability functions
1.4. Independent events
1.5. Conditional probability
2. Random Variables
2.1. Continuous and discrete random variable
Math 151. Rumbos
Fall 2013
1
Solutions to Review Problems for Exam 2
1
2 if 1 < x < ;
x
be the pdf of a random variable X . If E1
1. Let fX (x) =
0
if x 1,
denote the interval (1, 2) and E2 the interval (4, 5), compute Pr(E1 ), Pr(E2 ),
Pr(E1 E2 ) and Pr
Math 151. Rumbos
Spring 2014
1
Review Problems for Final Exam
1. Three cards are in a bag. One card is red on both sides. Another card is white
on both sides. The third card in red on one side and white on the other side. A
card is picked at random and pl
Math 151. Rumbos
Fall 2014
1
Solutions to Exam 2 (Part I)
1. Let X denote a discrete random variable with possible values x1 , x2 , . . . , xn .
(a) Let g denote a real valued function of a single variable. Give a formula for
computing the expectation E[g
Math 151. Rumbos
Fall 2014
Solutions to Assignment #7
1. Suppose the pdf of a random variable X is as follows:
4 (1 x3 ) for 0 < x < 1,
3
f (x) =
0
otherwise.
Sketch the pdf and determine the values of the following probabilities:
(a) Pr X <
(b) Pr
1
2
1
Math 151. Rumbos
Fall 2014
1
Solutions to Exam 1 (Part I)
1. Let (C, B, Pr) denote a probability space, and let A and B denote events in B.
(a) State what it means for A and B to be independent.
Answer: The events A and B are independent means that
Pr(A B
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #3
1. Consider two events A and B such that Pr(A) = 1/3 and Pr(B) = 1/2. Determine the value of Pr(B Ac ) for each of the following conditions:
(a) A and B are disjoint;
(b) A B;
(c) Pr(A B) = 1/8.
Solu
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #6
1. For each of the following, nd the value of the constant c for which the given
function, p(x), is the probability mass function (pmf) of some discrete random
variable.
(a) p(x) = c
2
3
x
, for x =
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #5
1. Let (C, B, Pr) be a probability space. Prove that if E1 and E2 are independent
c
events in B, then so are E1 and E2 .
Hint: Observe that E1 \ E2 is a subset of E1 .
Proof: Assume that E1 and E2 ar
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #4
1. Let (C, B, Pr) be a sample space. Suppose that E1 , E2 , E3 , . . . is a sequence of
events in B satisfying
E1 E2 E3 .
(1)
Then, lim Pr (En ) = Pr
Ek
n
.
k=1
Hint: Use the analogous result for an
Math 151. Rumbos
Fall 2014
1
Solutions to Assignment #2
1. Let A, B and C be subsets of a sample space C. Prove the following
(a) If A C and B C, then A B C.
(b) If C A and C B, then C A B.
Solution:
(a) Proof: If x A B, then either x A or x B. If x A the
Math 151. Rumbos
Fall 2014
Solutions to Assignment #9
1. Two discrete random variable, X and Y , are said to be independent if
Pr(X = x, Y = y) = Pr(X = x) Pr(Y = y)
for all possible values of x and y or X and Y , respectively.
Prove that if X and Y are d
Math 151. Rumbos
Fall 2014
Solutions to Assignment #8
1. Let X Uniform(a, b) and compute E(X).
Solution: Since X Uniform(a, b), its pdf is given by
1 , if a < x < b;
b a
fX (x) =
0,
otherwise.
Thus,
E(X) =
xfX (x) dx
b
=
a
=
x
dx
ba
x2
2(b a)
b
a
b 2 a2