MATH 464
HOMEWORK 7
SPRING 2013
The following assignment is to be turned in on
Thursday, March 28, 2013.
1. Consider the following experiment: Roll 2 fair, four sided dice. Consider
the following discrete random variables:
X = the number of odd dice.
Y =
MATH 464
HOMEWORK 1
SPRING 2016
The following assignment is to be turned in on
Thursday, January 28th, 2016.
1. Let A, B, and C be events (subsets) of a sample space .
Write each of the following events in terms of A, B, and C using intersections, unions,
> # hmwk5_r.txt
>
> # p is probability of heads
> p=0.25;
> X<-0
> Y<-0
> SquareX<-0
> SquareY<-0
>
> for (n in 1:100000) cfw_
+
+ # seq is used to record the flips
+ seq=as.character();
+
Math 464 - Fall 13 - Homework 4
1. We roll a six-sided die n times. Each time the die comes up 1, we ip a
fair coin. Let X be the number of heads we get. Note that the number of
times the coin is ipped is random. Find the mean and variance of X.
2. You ha
# hmwk5_r.txt
> # p is probability of heads
> p=0.25;
> X<-0
> Y<-0
> SquareX<-0
> SquareY<-0
>
> for (n in 1:100000) cfw_
+
# seq is used to record the flips
+
seq=as.character();
+
# flip until we get H
+
count1<-0;
+
got_head<-0;
+
while (got_head=0) c
Math 464 - Fall 13 - Homework 2
1. In the 1990s the ELISA test was used to test blood donations for the
AIDS virus. If a sample of blood has the virus, the test will be positive
99.9% of the time. If the sample does not have the virus the test will be
neg
6
Jointly continuous random variables
Again, we deviate from the order in the book for this chapter, so the subsections in this chapter do not correspond to those in the text.
6.1
Joint density functions
Recall that X is continuous if there is a function
5
Continuous random variables
We deviate from the order in the book for this chapter, so the subsections in
this chapter do not correspond to those in the text.
5.1
Densities of continuous random variable
Recall that in general a random variable X is a fu
3
3.1
Multiple Discrete Random Variables
Joint densities
Suppose we have a probability space (, F, P) and now we have two discrete
random variables X and Y on it. They have probability mass functions
fX (x) and fY (y). However, knowing these two functions
Final Exam Solutons - Math 464 - Fall 2011
1. I roll two tetrahedral (four-sided) dice. Let X be the number of dice that
show an odd number, Y the number that show an even number. So both X
and Y only take on the values 0, 1, 2.
(a) Find the joint probabi
4
Moment generating functions
Moment generating functions (mgf) are a very powerful computational tool.
They make certain computations much shorter. However, they are only a
computational tool. The mgf has no intrinsic meaning.
4.1
Denition and moments
De
2
Discrete Random Variables
Big picture: We have a probability space (, F, P). Let X be a real-valued
function on . Each time we do the experiment we get some outcome . We
can then evaluate the function on this outcome to get a real number X().
So X() is
MATH 464
HOMEWORK 2
SPRING 2013
The following assignment is to be turned in on
Thursday, January 31, 2013.
1. Suppose we pick a letter at random from the word MISSISSIPPI. Write
down a sample space and give the probability of each outcome?
2. In a group o
MATH 464
HOMEWORK 1
SPRING 2013
The following assignment is to be turned in on
Thursday, January 24th, 2013.
1. Let A, B, and C be events (subsets) of a sample space .
Write each of the following events in terms of A, B, and C using intersections, unions,
MATH 464:
TEST 2
MAKE UP
SPRING 2013
Name
I.D. Number
Question
Points Score
1
10
2
10
3
10
4
10
5
10
6
10
7
10
Total
70
1
2
SPRING 2013
Rules to the Make-Up:
Here are the rules to the make-up. You have two choices. Either you
turn in the make-up, or you d
MATH 464
HOMEWORK 8
SPRING 2013
The following assignment is to be turned in on
Thursday, April 4, 2013.
1. Let X be a Poisson random variable with parameter > 0.
a) Find the moment generating function for X.
b) Use your result above to nd the mean of the
MATH 464
HOMEWORK 5
SPRING 2013
The following assignment is to be turned in on
Thursday, February 21, 2013.
1. Let x, y R and take n 2 an integer. Prove that
n
(x + y)n =
k=0
n k nk
x y
k
using an induction argument.
2. Let X be a binomial random variable
MATH 464
HOMEWORK 6
SPRING 2013
The following assignment is to be turned in on
Thursday, March 7, 2013.
1. Suppose that in a certain state the license plates have three letters followed by 3 numbers. If no letter or number can be repeated, how many
licens
MATH 464
HOMEWORK 4
SPRING 2013
The following assignment is to be turned in on
Thursday, February 14, 2013.
1. Let X be a discrete random variable on a probability space (, F, P ).
Let g : R R be a function and set Y = g(X), i.e. Y : R is dened by
Y () =
MATH 464
HOMEWORK 3
SPRING 2013
The following assignment is to be turned in on
Thursday, February 7, 2013.
1. Three couples are invited to a dinner party. They will independently
show up with probabilities 0.9, 0.8, and 0.75 respectively. Let N be the
num
1
Probabilities
1.1
Experiments with randomness
We will use the term experiment in a very general way to refer to some process
that produces a random outcome.
Examples: (Ask class for some rst)
Here are some discrete examples:
roll a die
ip a coin
ip a
A
Counting
A.1
First principles
If the sample space is nite and the outomes are equally likely, then the probability
measure is given by P (E) = |E|/| where |E| denotes the number of outcomes in the
event E. So to compute probabilities in this setting we
HW 3 Math 464
Due in class Friday, October 7, 2016.
1. An urn contains 4 balls, each of a different color. In succesive draws, draw a ball and
note its color, and then put it back in the urn. Let N be the number of draws until each
of the 4 colors have be
Homework 1
Due September 6
Problems 1.20, 1.26, 1.30 from the textbook
Add-on sketch of a proof of existence of a set which is not Lebesgue-measurable. The original example is due to G. Vitali. Let = [0, 1); we will show that there is no translation-invar
Sample Exam 2 - Math 464 - Fall 11 -Kennedy The questions on this sample exam are meant to be representative of the questions that will be on the exam. However, if a topic does not appear on the sample exam that does not mean it will not appear on the exa
Sample Exam 2 Solutions - Math 464 - Fall 10 -Kennedy 1. Let X have a gamma distribution with = 2, w = 3. Let Y = 3X. Show that Y has a gamma distribution and find the values of and w for Y . (Hint: how is the moment generating function of Y related to th