M. Bremer
Math 163 - Fall 2015
Limit Theorems
Recall, that in the very beginning of the course, probability was defined in the
frequentist sense as the limiting long-run frequency of the occurrence of an event.
We now want to make this idea more precise.
Exam II - Solutions for Review Problems
Math 163 - Fall 2015
Note: There is no guarantee for the correctness of these answers. If you think you found a
mistake, please let me know! Below, nd the answers for the Math 161A review problems
and the actuarial
Exam I - Solutions for Review Problems
Math 163 - Fall 2015
Note: There is no guarantee for the correctness of these answers. If you think you found
a mistake, please let me know!
1. (A B c ) (B Ac )
B Ac C c
(A B) (B C)
(A B) (B C)
2. (a) 0.2
(b) 0.5
3.
Math 163 - Fall 2015
Quiz 4 - Version a
Your Name:
1. For the following situations, identify the name of the distribution of X and the
values of all parameters of the distribution.
(a) (5 points) X is the number of hearts in a poker hand (5 cards dealt fr
Solution for Quiz 5 - Version b
Math 163 - Fall 2015
Recall, that the area of a circle with radius r is r2 .
Let X and Y be continuous random variables with joint probability density
f (x, y) =
1
4
1 < x < 1, 1 < y < 1
otherwise
0
(a) (10 points) Find P (
Quiz 2 - Version b
Math 163 - Fall 2015
Your Name:
A survey of 100 families found that 57 families had a dog. 55 of the families either
said that they had a cat, or they said that they had no dog. How many families had
both a cat and a dog?
Dene any abbre
Math 163 - Fall 2015
Quiz 3 - Version a
Your Name:
Let X be a continuous random variable with probability density function
1/3 1 x 2
2/3 3 x 4
f (x) =
0
otherwise
(a) (8 points) Draw a labeled sketch of f (x) (dont forget to label your axes).
(b) (6 poi
Quiz 2 - Version a
Math 163 - Fall 2015
Your Name:
Suppose that of 100 ve year olds, 40 know how to swim. 75 of the children either
do not know how to swim or do not know how to ride a bike. How many children
know how to swim but do not know how to ride a
Solution for Quiz 7 - Version b
Math 163 - Fall 2015
Consider two discrete random variables X and
shown below.
X
1
0
3/8 1/8
Y 1
1
0 1/8
Total 3/8 2/8
Y with joint probability mass function
1
0
3/8
3/8
Total
1/2
1/2
1
(a) (10 points) Find Cov(X, Y ).
Firs
Martina Bremer - SJSU
Name
Uniform
Exponential
Named Continuous Distributions
PDF
CDF
E(X) V (X) Parameters
x<a
0
1
(ba)2
xa
a+b
ba a x b
axb
F (x) =
f (x) =
a, b endpoints
2
12
0
otherwise
ba
1
x>b
Example: A bus is going to arrive at the bus stop some
Exam II - Solutions for Review Problems
Math 163 - Fall 2015
Note: There is no guarantee for the correctness of these answers. If you think you found a
mistake, please let me know! Below, nd the answers for the Math 161A review problems
and the actuarial
Math 163 - Fall 2015
Homework 9
1. The joint density function of X and Y is
x + y 0 < x < 1, 0 < y < 1
f (x, y) =
0
otherwise
(a) Are X and Y independent?
(b) Find the density function of X.
(c) Compute P (X + Y < 1).
(d) Compute P (X < Y 2 ).
2. Let U de
M. Bremer
Math 163 - Fall 2015
Combinatorial Analysis
Combinatorics is the science of counting. Usually, we count in how many ways
something specific can happen to compute probabilities. Since combinatorics were
introduced quite thoroughly in your introdu
Midterm II - Review
Math 163 - Fall 2015
The second midterm in Math 163 will concentrate on the topics listed below. However,
probability classes tend to be sequential. That means that while the focus will be on
the chapters covered since the rst midterm,
PROBABILITY THEORY
1. Prove that, if A and B are two events, then the probability that at least
one of them will occur is given by
P (A B) = P (A) + P (B) P (A B).
China plates that have been red in a kiln have a probability P (C) =
1/10 of being cracked,
Midterm I - Review
Math 163 - Fall 2015
For the rst midterm exam in Math 163 you should be familiar with the following topics:
Combinatorics
You should be able to solve counting problems of intermediate diculty in a reasonable time-frame. The best way to
Homework 9
Math 163 - Fall 2015
1. The joint density function of X and Y is
x + y 0 < x < 1, 0 < y < 1
0
otherwise
f (x, y) =
(a) Are X and Y independent?
No, since the density function does not factor into parts depending only on x
and y, respectively.
(
Solution for Homework 8
Math 163 - Fall 2015
1. Let X 2 be a random variable with a 2 -distribution with n degrees of freedom.
n
(a) Derive E[X k ] for general k > n/2.
k
k
E[X ] =
x fX (x)dx =
0
n
x 2 1 x
x n
e 2 dx =
22 n
2
k
0
0
n
xk+ 2 1 x
e 2 dx
n
22
Solution for Homework 6
Math 163 - Fall 2015
1. Suppose that it takes at least 9 votes from a 12-member jury to convict a defendant.
Suppose also that the probability that a juror votes a guilty person innocent is 0.2,
whereas the probability that the jur
Math 163 - Fall 2015
1. Let X
2
n
Homework 8
2 -distribution
be a random variable with a
(a) Derive E[X k ] for general k >
with n degrees of freedom.
n/2.
(b) In particular, nd E[1/X].
(Hint: Instead of evaluating an integral of the form
1
Z
e
x r 1
x
d
Solution for Quiz 5 - Version a
Math 163 - Fall 2015
Recall, that the area of a circle with radius r is r2 .
Let X and Y be continuous random variables with joint probability density
f (x, y) =
1
4
0 < x < 2, 0 < y < 2
otherwise
0
(a) (10 points) Find P (
Solution for Quiz 4 - Version b
Math 163 - Fall 2015
1. For the following situations, identify the name of the distribution of X and the values
of all parameters of the distribution.
(a) (5 points) X is the number of aces in a poker hand (5 cards dealt fr
Solution for Quiz 3 - Version a
Math 163 - Fall 2015
Let X be a continuous random variable with probability density function
1/3 1 x 2
2/3 3 x 4
f (x) =
0
otherwise
(a) (8 points) Draw a labeled sketch of f (x) (dont forget to label your axes).
(b) (6 po
Homework 5
Math 163 - Fall 2015
1. Three fair six-sided dice are tossed. Let X be the number of distinct faces shown
(e.g., if you roll 1,5,1 there are two disctinct faces).
(a) Write down the probability mass function of X.
(b) Find the expected value an
Homework 4
Math 163 - Fall 2015
1. For each of the following statements, decide whether they are always true (i.e., true
for all sets A, B, C) or not. If they are not always true, provide a counter example
for instance in the form of a Venn Diagram or by
Math 163 - Fall 2015
M. Bremer
Limit Theorems
Recall, that in the very beginning of the course, probability was dened in the
frequentist sense as the limiting long-run frequency of the occurrence of an event.
We now want to make this idea more precise. To
M. Bremer
Math 163 - Fall 2015
Random Variables
Denition: A random variable X is a real valued function that maps a sample
space S into the space of real numbers R.
X:SR
As such, a random variable summarizes the outcome of an experiment in numerical
form.
M. Bremer
Math 163 - Fall 2015
Joint Distributions
So far, we have studied probability models for a single random variable. A named distribution, for example, can be used to compute probabilities, averages, or variances
for a single X. In most application
M. Bremer
Math 163 - Fall 2015
Conditional Probability and Independence
The concept of conditional probability is immensely useful. It allows to connect
events that may have already happend with events that have not yet happened and
makes it possible to m
M. Bremer
Math 163 - Fall 2015
Properties of Expectation
We have already derived a few properties of expected values earlier in this course.
Recall, that the expected value of a discrete random variable X with PMF p(x) is
dened as
E[X] =
xp(x)
x:p(x)>0
wh