Homework 11
Math 163 - Fall 2015
1. A die is continually rolled until the total sum of all rolls exceeds 300. Approximate
the probability that at least 80 rolls are necessary.
2. How often do you have
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
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
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 )
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
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
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
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 sketc
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
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/
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
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
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 answe
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
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 combina
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
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 intermedi
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 facto
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
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 perso
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: Instea
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) Compu
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 answe
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 sket
Quiz 1 b
Math 163 - Fall 2015
In both problems below, state your answer as an integer (not as an expression involving
factorials).
1. 8 professors, the department chair and the dean are seated in a ro
Solution for Quiz 4 - Version a
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
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, ave
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 event
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 variab
M. Bremer
Math 163 - Fall 2015
Axioms of Probability
Denition: The set of all possible outcomes of an experiment is called the sample
space. The possible outcomes themselves are called elementary even