AMS412
Q1:
Prof. Wei Zhu
, and the value of random variable W depends on a coin (fair coin) flip:
, find the distribution of W. Is the joint distribution of Z and W a
bivariate normal?
A1:
So
.
The jo
AMS412.01
Homework 1
Spring 2014
1. Jake has been caught stealing cattle, and is brought into town for justice. The judge
is his ex-wife Gretchen, who wants to show him some sympathy, but the law cle
AMS412
HomeWork # 3. Solutions
Prof. Wei Zhu
1.
The gunner on a small assault boat fires three missiles at an attacking plane. Each has
a 20% chance of being on target. If two or more of the shells fi
AMS412.01
Homework 1
Spring 2014
1. Jake has been caught stealing cattle, and is brought into town for justice. The judge is his
ex-wife Gretchen, who wants to show him some sympathy, but the law cle
Quiz 1. Solutions
Question. If X ~ N( , 2 ), what is its mgf M X (t ) ?
Solution:
1
( x )2
M X (t ) = E (e ) = exp( xt )
exp(
)dx
2 2
2
tX
=
=
1
2 2 xt + x 2 2 x + 2
exp(
)dx
2 2
2
1
x 2 2( 2t ) x +
Problem Set 3 Due July 30
ECON 139/239
2010 Summer Term II
1. Multiple Choice. Explain your response briefly.
(a) Which of the following regressions suffer from perfect collinearity?
a. wage = 0 + 1 m
AMS 412
Professor Wei Zhu
Jan 29th
1. Review of Probability, the Monty Hall Problem
(http:/en.wikipedia.org/wiki/Monty_Hall_problem)
The Monty Hall problem is a probability puzzle loosely based on the
AMS412.01
Homework 4
Spring 2015
Name: _ ID: _ Signature: _
Instruction: Dear students, this homework is due before class on Thursday, 2/26/2015.
i .i .d .
i .i .d .
2
1. Let X1 , , X n ~ N ( 1 , 12
Quiz 4 Solutions
i .i . d .
Y1 , , Yn ~ U [ , 0]
(a) Find the MOME for
(b) Find the MLE for
(c) Are the MOME and MLE unbiased estimators of ?
Solution:
1
(a) f ( y ) = , y 0
0
1
y2 0
1
E (Y ) = y dy
P a g e |1
Exercises 2.4
2.4.11
(a) Let C=colorblind, M=male, and F=female
P (C ) = P ( M ) P (C | M ) + P ( F ) P (C | F )
= .45(.06) + .55(.075)
= .027 + .004125
= .031125
(b) P (C | M ) = .06
2.4.1
One and Two-sample t-tests
The R function t.test() can be used to perform both one and two sample t-tests on
vectors of data.
The function contains a variety of options and can be called as follows:
>
Significance Testing Using R
In the following handout words and symbols in bold indicate R functions and words and
symbols in italics indicate entries supplied by the user; underlined words and symbol
AMS412.01
Spring 2015
Practice Midterm
Name: _ ID: _ Signature: _
Instruction: This is a close book exam except for an 8x11 cheat sheet (double sided). Anyone who cheats in
the exam shall receive a g
Quiz 6.
1. Let
be a random sample from a normal population N(
(a) Derive the distribution of
). Please
=
(b) Derive the distribution of
(c) Derive the distribution of
, where S is the sample standard
Quiz 2. Solutions
1. Linear transformation : Let X ~ N ( , 2 ) and Y a X b , where a&b
are constants, what is the distribution of Y?
Solution:
M Y (t ) E(e tY ) E[e t ( aX b) ] E(e atX bt ) E(e atX e
Other Common Univariate Distributions
Dear students, besides the Normal, Bernoulli and Binomial distributions, the
following distributions are also very important in our studies.
1. Discrete Distribut
1
HW#6.
5.4.3, 5.4.4, 5.4.9,
5.5.1, 5.5.2, 5.5.3, 5.5.4, 5.5.5, 5.5.6,
5.5.11, 5.5.15, 5.5.21, 5.5.22, 5.5.30, 5.5.32, 5.5.41
5.4.3
(a) Z is a standard normal random variable with known variance, thus
AMS412
HomeWork # 3. Solutions
Prof. Wei Zhu
1.
The gunner on a small assault boat fires three missiles at an attacking plane. Each has
a 20% chance of being on target. If two or more of the shells fi
Inference on two population means (and two population variances)
1. The samples are paired
paired samples t-test
2. The samples are independent independent samples t-test
a)
12 2 2 pooled-variance t-t
Point Estimators, MLE & MOME
Point Estimators
Example 1. Let X1, X2, ,
be a random sample from N(
Please find a good point estimator for 1.
).
2.
Solutions. 1.
2.
There are the typical estimators for
Lecture 10. Inference on one population mean &
the Exact Confidence Interval for
when the population is normal &
is known
Motivation & simple random sample
Eg) We wish to estimate the average height
AMS412
Lecture 9. Cramer-Rao Lower Bound, Efficient Estimator,
Best Estimator
Unbiased Estimator of , say
( ) when there are many of them.
It could be really difficult for us to compare
Theorem. Cram
Confidence Interval, continued; & related sample size calculations
1. Sample size estimation based on the large sample C.I. for p
From the interval p Z 2
p(1 p)
, p Z 2
n
p(1 p)
n
L lengh of your 1
Inference on One Population Mean
Hypothesis Testing
Scenario 1. When the population is normal, and the population variance is
known
Data : X 1 , X 2 , X n
i .i . d .
~ N ( ,
2
)
H 0 : 0
H 0 : 0
H
1. Scenario 1: Review of Confidence Interval for One Population Mean when the
Population is Normal and the Population Variance is Known
Dear students, before we study the new topic today: CLT and the
Hypothesis Test on One Population Mean (continued)
Scenario 1: When the population is normal, and the population variance
known
Scenario 2: Any population (usually not normal), but the sample size is
AMS412 Lecture Notes #2
Review of Probability (continued)
Probability distributions.
(1)
Binomial distribution
Binomial Experiment:
1) It consists of n trials
2) Each trial results in 1 of 2 possible
Power of the test & Likelihood Ratio Test
Power Calculation (Inference on one population mean)
Likelihood Ratio Test (one population mean, normal
population)
Truth
H0
Decision
Ha
Type II error
H0
Type
t-Test Statistics
Overview of Statistical Tests
Assumption: Testing for Normality
The Students t-distribution
Inference about one mean (one sample t-test)
Inference about two means (two sample t-test)