7.1) Let Xl,..., Xn be a sample from the distribution whose density function is
-(x-9)
x) 2 {e x 2 0
0 otherwise
Determine the maximum likelihood estimator of 9.
f(x1,...,xn l6) : ((149) 2 e H
Therefore log[f(xl,...,xn |0)] =:xi ~0=n0:xi
i=l [:1
Thi
1
Review of Set Theory
To build a probability model we usually have an experiment in mind and are interested in assigning probabilities to certain events of interest. For example, the experiment may be to roll two dice and an event of
interest may be that
Stat-IEOR 4150: Probability and Statistics
Midterm 2010
Professor Guillermo Gallego
INSTRUCTIONS:
1. The exam is closed book and closed notes. You are allowed to use a calculator and one 8.5
by 11 sheet with formulas, both sides allowed. This is a 75 minu
Midterm Review Session
Descriptive Statistics
1. Sample Mean, Sample Median, and Sample Mode
n
Sample Mean x = i=1
n
Sample Median (P 20)
Sample Mode (P 22)
xi
2. Sample Variance and Sample Standard Deviation
n
(xi )2
x
(n1)
Sample Variance s2 =
i=1
n
Sam
1. Consider the experiment of rolling two fair dice and observing the resulting
value. (Assume the dice has six faces labled 1, . . . , 6.)
(a) Write down the sample space.
Answer: The sample space is = cfw_(m, n) : m, n cfw_1, . . . , 6 so has
it has 36
1
Classical Methods of Estimation
A point estimate of some population parameter is a single valued of a statistic . It is useful to distinguish
between the estimator (a random variable), the point estimate and the (unknown parameter) . An
estimator is sai
Stat-IEOR 4150: Probability and Statistics
Midterm 2009
Professor Guillermo Gallego
INSTRUCTIONS:
1. The exam is closed book and closed notes. You are allowed to use a calculator and one 8.5
by 11 sheet with formulas, both sides allowed. This is a 75 minu
1. Let X and Y have the joint probability mass function given by the following table:
X/Y
0
1
0
0
1/8
1
1/8
1/4
2
1/4
1/8
3
1/8
0
(a) Find the probability mass function of X.
Answer: By adding the rows we nd that px (0) = px (1) = 1/2.
(b) Find the probab
Stat-IEOR 4150: Probability and Statistics
Midterm 2009
Professor Guillermo Gallego
INSTRUCTIONS:
1. The exam is closed book and closed notes. You are allowed to use a calculator and one 8.5
by 11 sheet with formulas, both sides allowed. This is a 75 minu
IEOR W4150
Dr. A. B. Dieker
Introduction to Probability and Statistics
Fall 2016
Homework 8
due on Wednesday November 2, 2016, 4:10pm EST
Include all intermediate steps of the computations in your answers. If the answer is readily
available on the web (e.
IEOR W4150
Dr. A. B. Dieker
Introduction to Probability and Statistics
Fall 2016
Homework 4
due on Wednesday October 5, 2016, 4:10pm EST
Include all intermediate steps of the computations in your answers. If the answer is readily
available on the web (e.g
IEOR W4150
Dr. A. B. Dieker
Introduction to Probability and Statistics
Fall 2016
Homework 5
due on Wednesday October 12, 2016, 4:10pm EST
Include all intermediate steps of the computations in your answers. If the answer is readily
available on the web (e.
IEOR W4150
Dr. A. B. Dieker
Introduction to Probability and Statistics
Fall 2016
Homework 10
due on Wednesday November 23, 2016, 4:10pm EST
Include all intermediate steps of the computations in your answers. If the answer is readily
available on the web (
IEOR W4150
Dr. A. B. Dieker
Introduction to Probability and Statistics
Fall 2016
Homework 9
due on Wednesday November 9, 2016, 4:10pm EST
Include all intermediate steps of the computations in your answers. If the answer is readily
available on the web (e.
IEOR W4150
Dr. A. B. Dieker
Introduction to Probability and Statistics
Fall 2016
Homework 3
due on Wednesday September 28, 2016, 4:10pm EST
Include all intermediate steps of the computations in your answers. If the answer is readily
available on the web (
IEOR W4150
Dr. A. B. Dieker
Introduction to Probability and Statistics
Fall 2016
Homework 6
This homework does not have to be turned in.
1. Suppose the continuous random variable U has a uniform distribution on (1, 1). Show that U
and U 2 are not independ