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Markov's Inequality and LLN for Sample Means
Markov's Inequality
Let X be a continuous random variable with PDF fX(x), E(X) = X and P(X > 0) = 1. Then
c
c
X = 0 x fX(x) dx 0 x fX(x) dx c0 fX(x) dx = cP(X c), for c > 0.
Thus P(X > c) X/c and P(X < c
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Matching in Sequences for Values and Indexes
Two vectors
Consider two numerical vectors of length 10: 1:10 and
ten = c(1,3,5,7,9,2,4,6,8,10)
Logical equality (vectors of equal length)
ten = 1:10
[1] TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FA
Wedad Sadun:NETID:fw8234.
In terms of problem (8):
We have x=cfw_-1,0,1,2, so it is clear that the product is going
to be 0 as long as 0 is an element of this set.
> a=3
> x1=-1:2
> x2=rep(1:0,2)
> sum(a*x1)
[1] 6
> prod(x1^a)
[1] 0
In terms of problem 12
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How Many Shuffles to Randomize a Deck of Cards?
In a friendly card game, many people shuffle the deck of cards only two or three times between
hands. A statistical analysis of shuffling shows this is not enough to put cards in anything close to
a r
STAT 6304 HOMEWORK SIX
1
STAT 6304 HOMEWORK SIX
2
STAT 6304 HOMEWORK SIX
3
STAT 6304 HOMEWORK SIX
4
STAT 6304 HOMEWORK SIX
5
STAT 6304 HOMEWORK SIX
6
STAT 6304 HOMEWORK SIX
7
STAT 6304 HOMEWORK SIX
8
STAT 6304 HOMEWORK SIX
9
STAT 6304 HOMEWORK SIX
10
STAT
# Kernel estimate of f
n <- 100
x <- rchisq(n,8)
#h <- 1
h <- c(0.5, 1, 5)
t <- seq(0,25, length=n)
fhat1 <- matrix(0, nrow=n, ncol=length(h)
plot(t, dchisq(t,8), type='l')
for (j in 1:length(h)cfw_
for (i in 1:n)cfw_
fhat1[i,j] = 1/(n*h[j])*sum(dnorm(t[i
Course 6204: Probability Theory
Fall Quarter 2015
Instructor Name: Dr. Ayona Chatterjee
Email: [email protected]
Office: SC N-319 (Hayward campus)
Office Hours: Tuesday 6:00-7:00pm and
Thursday 3:00 4:00 pm and 6:00-7:00pm.
Any change in off
Homework 3 Solutions
1. Find the m.l.e. of based on a random sample X1 , X2 , . . . , Xn from
each of the following p.d.f.s.
(a)
f (x|) = x1 ,
0 < x < 1, 0 <
n
X
`() = n ln() + ( 1)
ln(xi )
i=1
d`()
n
= +
d
and
So
and
n
X
ln(xi )
i=1
n
d2 `()
= 2
2
d
n
Homework 2 Solutions
8.5
a. 1 = E(X) = (1) + (2)(1 ) = 2 . Solving for , we get =
2 1 . Plugging in the sample moment we get the method of moments
estimator
= 2
1 = 2 X.
From the data, we have = 2 5 = 1 .
3
b. L() =
Q3
i=1
3
[(3 2xi ) + xi 1] = (1 )2
c.
Associate Professor J. Kerr
Final Exam
STAT 6501
Fall 2015
Instructions: You have 110 minutes to complete the exam. Please write all solutions in
your greenbook starting each new problem on a new page, including all pertinent work for
full credit. Partial
Associate Professor J. Kerr
Midterm II
STAT 6501
Fall 2016
Instructions: You have 110 minutes to complete the exam. Please write all solutions in
your bluebook starting each new problem on a new page, including all pertinent work for
full credit. Partial
Homework 1 Solutions
7.3
a. parameter
b. parameter
c. fixed
d. random variable
e. parameter
f. random variable
g. parameter
h. random variable
7.5
The sample mean is a random variable before a sample is taken since the
sample is assumed to be drawn random
Associate Professor J. Kerr
Midterm I
STAT 6501
Fall 2016
Instructions: You have 110 minutes to complete the exam. Please write all solutions in
your Green Book starting each new problem on a new page, including all pertinent work for
full credit. Partial
# read in cancer data
cancer <read.csv(url("http:/statistics.csueastbay.edu/~jkerr/STAT65012/cancer.txt"),
header=FALSE, col.names=c("mortality","female")
N <- length(cancer$mortality)
#part a
graphics.off()
x11()
hist(cancer$mortality, freq=FALSE, main='
Associate Professor J. Kerr
Final Exam
STAT 6501
Fall 2015
Instructions: You have 110 minutes to complete the exam. Please write all solutions in
your greenbook starting each new problem on a new page, including all pertinent work for
full credit. Partial
Smith 1
Philip Smith
Professor Marina Sapozhnikov
English 3003
24 October 2016
Essay 1: Draft 2
Performance coaching and positive psychology are two fields that are more closely
related than most people think. These fields are discussed in Angela R. Mouto
Study Guide
Chapter 6
Operant Conditioning: Comes from operant response; behavior operates/mechanism on
the environment to get reinforcement/punishment.
- EX: Person makes dinner (operant response) to be able to eat/receive social praise or
compliments on
Shushan Tekeste
October 11, 2016
Dr. Asha Rao
Syllabus Question
1. List your top 5 motivation at work. What factors motivate you to work hard, be persistent, and
work toward the objectives of the organization? They can be internal or external in the work
Philip Smith
Psych 4390
10/24/16
Analysis of Mind Freud Reaction
This documentary was quite interesting to me. It details Sigmund Freuds life in a well
thought out and chronological manner. I truly appreciated being able to learn about his early
childhood
Prasad 1
Shoma Prasad
Instructor Ingrid
English 1001
03 November 2016
Injustice System of America: U.S System of Segregation
Some people might think an era of slavery has disappeared but, there is still ideology of
slavery that still exist today. Mistreat
Worksheet #1: Developing a Thesis
1. Examine your previous response:
What was your initial response to In-Class Essay 1 prompt question (Do you
believe that the government should forgive student loans thus making the
taxpayers bear the burden, or do you t
Running head: REFLECTIONS ON SESSION ONE
Putting CBT into Practice: Reflections on session one
Amber Kimmins
SW 6505: Advanced Micro Practice
1
REFLECTIONS ON SESSION ONE
2
My first session with CJ was interesting. CJ talks very fast and describes herself
Cosby, Javhan
Psych 4210
Dr. Marvin Lamb
Week 4
1) Marshalls conduct in school gets him in a lot of trouble and his mother does not know
what to do in order to change this. If Skinner could speak with his mother, what advice
would he give her?
He would ta
All people have different capacities and not all will be equal or all have the same advantage if we all
share separate interest.
Men who are charmed by democracy are charmed by equality and equality will subdue the urge for
freedom according to Tocquevill
October 17, 2016
STAT 3031: Midterm No. 1
Page 1 of 12
California State University East Bay
Prof. R. Martnez
Instructions
The exam covers the material covered in class from Chapters 1, 2, 3 of
the textbook and contains 8 problems for a total of 126 points
November 7, 2016
STAT 3031: Midterm No. 2
Page 1 of 9
California State University East Bay
Prof. R. Martnez
Instructions
The exam covers the material covered in class from Chapters 4 and 5 of
the textbook and contains 8 problems for a total of 125 points.