HW 11 (10 points), due on July 1st, Friday
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You must do HW by yourself and do not copy others' solutions
1. A manager evaluates eectiveness of a major hardware upgrade by running
a certa
Poisson Distribution
This is a distribution useful for modeling of the number of events occuring in an interval of space or time,
for example, the number of automobiles arriving at a toll booth during a 10-minute period or the number of
weak points in a s
Goodness of t (Contd)
Review: sample correlation as a measure of goodness of t
Second measure of goodness of t: Coecient of determination R2, it is
based on a comparison of variation accounted for by the line versus raw
variation of y .
Ideas: The quantit
Testing Hypothesis (Contd)
Recall: In the previous lecture, 0.021 is called the p-value for testing the
null hypothesis p = 0.5 against an alternative hypothesis p = 0.5.
Formal procedure: Let
H0 : p = 0.5 v.s. H1 : p = 0.5
then the test statistic is T =
Examples for CI of proportion
Example 1: Suppose we want to estimate the fraction of records in the
2000 IRS data base that have a taxable income over 35K.
Question: We want to get a 98% condence interval and wish to estimate
the quantity to within 0.01.
Example for MLE:
Review: What is MLE? How to nd it (5 steps)?
may be multiple: Rp with p > 1
Example: Let X1, . . . , Xn be i.i.d N (, 2), both and 2 are unknown.
x1, , xn are the data/sample value of X1, , Xn
What is the pdf of normal random variable?
Estimator (Contd)
Review: What is estimator? What is estimates? What are the properties
we used to compare estimators?
Example: The sample mean x is consistent for . That means that if the
sample size is getting large, then X is getting very closed to in
Other descriptive statistics
Review: Descriptive statistics, inferential statistics, sample/population
mean, sample/population variance, sample/population median, range
Population quantile: A p-quantile of a population is a number x that solves
equations
Statistical inference
Question: What is Statistics?
Statistics: is the science and art of studying data. It involves collecting,
classifying, summarizing, organizing, analyzing, and interpreting numerical
information.
More: In general, there are two diere
The M/M/c queue
Again, X (t) the number of individuals in the queueing system can be
modeled as a birth & death process.
The transition state diagram for the X (t) is:
0
2
1
2
K
c-1
3
(c-1)
1
c
c
c
Clearly, the critical thing here in terms of whether or n
The M/M/1 Queue: Example
Printer Queue (continued) A certain printer in the Stat Lab gets jobs with
a rate of 3 per hour. On average, the printer needs 15 min to nish a job.
Let X (t) be the number of jobs in the printer and its queue at time t. We
know a
Queuing Queueing system
systems
server 1
enter the system
some population of
individuals
server 2
according to some
random mechanism
exit the
system
server c
Depending upon the specic application there are many varieties of queuing
systems:
size & nature
Random Number Generation
1
Introduction
Anyone who considers arithmetical methods of producing random digits is, of course, in a
state of sin. This famous statement concerning the use of sequences of numbers generated
using recursive formulas as random se
Generating Random Numbers: Some Examples
Example 1:
The following 10 numbers are realizations from a Standard Uniform distribution:
.54463 .15389 .85961 .61149 .05219 .41417 .28357 .17783 .40950 .82995
Explain how to use these numbers to generate 10 iid B
HW 10 (10 points), due on June 24th, Friday
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You must do HW by yourself and do not copy others' solutions
1. The Weibull distribution has two parameters and , its density function
f (x)
HW 9 (10 points), due on June 21st, Tuesday
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You must do HW by yourself and do not copy others' solutions
1. 8.2, 8.8 from Baron's book
8.2 from Baron's book:
(a) Using R, or SAS, or Ma
HW 8 (10 points), due on June 17th, Friday
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You must do HW by yourself and do not copy others' solutions
1. A bank has two agents that helps the customers and a waiting hall that
holds a
HW 7 (10 points), due on June 14th, Tuesday
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You must do HW by yourself and do not copy others' solutions
1. 6.4, 6.6, 6.7 from Baron's Book
6.4 from Baron's Book:
(a) Let x(n) = 1 if t
HW 6 (10 points), due on June 8th, Wednesday
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You must do HW by yourself and do not copy others' solutions
1. The price of a particular make of a 32GB iPad among dealers nationwide is
assu
HW 5 (10 points), due on June 3rd , Friday
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You must do HW by yourself and do not copy others' solutions
1. 3.33, 4.5, 4.11 from Baron's book
3.33 from Baron's book: The number of the l
HW 4 (10 points), due on May 31st, Tuesday
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You must do HW by yourself and do not copy others' solutions
1. Let X be a random variable with image Im(X ) = cfw_0, 1, 2, 3,
(a) Fill in the
HW 3 (10 points), due on May 27th, Friday
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You must do HW by yourself and do not copy others' solutions
1. Answer the following two problems:
(a) Plants A, B, C produce 35%, 15% and 50%
HW 2 (10 points), due on May 24th, Tuesday
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You must do HW by yourself and do not copy others' solutions
1. 2.6, 2.14 from Baron's book
2.6 from Baron's book: Under good weather conditi
HW 1 Solution (10 points), due on May 20th, Friday
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1. A coin is tossed three times, and the sequence of heads and tails is
r
Suppose S = cfw_X1 , . . . , Xn is a simple random sample from a population with and nite variance
2 < . Show that the sample mean X is an unbiased estimator for , so is sample variance s2 for 2 .
1. Before moving to formal proofs, there are several pro
Balance equations:
Balance equations: In the context of physical-chemistry, it is called the
master equation.
Flows: The Flow-In = Flow-Out Principle provides us with the means to
derive equations between the steady state probabilities.
State 0:
1 p1 = 0
Birth and Death Processes
Review: What is a stochastic process? What is a Poisson Process?
Motivation: Birth and Death process (B + D) is a generalization of Poisson
process, and it provides for modeling of queues, i.e. we assume that arrivals
stay some t
Friday, 21st
1. Review:
Probability:
Independence:
Counting: Two principles, Permutation (ordered sample) with replacement, without replacement
Example: A survey question lists seven pizza toppings and ask you
to rank your favorite 3. How many possible an
Wednesday, 19th
1. Review:
Concept: Uncertainty/randomness, probability, mathematical model
(random experiment, outcome , sample space , event E).
Set theory: set, operations (belongs, subset, empty set, , , A, A \
B ), disjoint, exhaustive, De Morgan's l