STAT 135 Lab 3 solutions
February 9, 2015
Exercise 1: MLE: Asymptotic results
In class, you showed that if we have a sample Xi P oisson(0 ), the MLE of is
1
M L = X n =
n
n
Xi
i=1
1. What is the asymptotic distribution of M L (You will need to calculate t
Stat 135 Section 2
Christine Ho
September 9, 2015
Estimating population mean and variance: Question 35
Introduction to simulation
Simulating the sampling distribution of the mean
Reproducing Figure 7.2 (more advanced R coding)
Putting it all together
Stat 135, Fall 2011
A. Adhikari
HOMEWORK 5 (due Friday 10/7)
1. 9.11. Use R to do the plots accurately. In each case, say what the limiting power is as approaches
0.
2. 9.12.
3. 9.13. In c, use R to nd the critical values x0 and x1 . Turn the page for d.
Stat 135, Fall 2011
A. Adhikari
HOMEWORK 4 (due Friday 9/30)
1. 8.20. Use R to do this one accurately. Be careful about constants when youre working out the
distribution of 2 .
2. 8.32. You dont have to do all six intervals in parts (b) and (c); just do t
Stat 135, Fall 2011
HOMEWORK 8
due WEDNESDAY 11/9 at the beginning of lecture
Friday 11/11 is a holiday, so this is a very short homework due two days earlier at the
start of lecture.
Grading: A (4 points) for all three problems done well, B (2 points) fo
STAT135 Spring 2015
hw#3 solutions
February 17, 2015
8.28 (2 points)
The intervals in the left panel of Figure 8.8 have dierent centers because the centers correspond to = X ,
which is a random variable that can take on dierent values each time it is obse
Lab 2 Solutions Stat 135
Tessa Childers-Day March 3, 2011
1
Chapter 8, Problem 43
Please note that I was very specific of the format of the report. I posted an example report, and noted several times that the code was to be printed separately from the ans
Stat 135, Fall 2011
HOMEWORK 1 (due Friday 9/9)
1a) Let x1 , x2 , . . . , xn be a list of numbers with mean and SD . Show that
2 =
1
n
n
x2 2
i
i=1
b) A class has two sections. Students in Section 1 have an average score of 75 with an SD of 8.
Students in
Stat 135 Fall 2014 Lec 2
1. (a) z/2 s = 1.96
p
(b)
i.
ii.
iii.
iv.
v.
0.14.0.86
10171
MIDTERM EXAM SOLUTIONS
= 0.02133
False. Almost same FPC. FPC for CA = 0.999973, FPC for FL = 0.999946, hence False.
False. Need to quadraple.
False. Either in or out, so
HW # 1 Solutions
(all problems from Nolan/Speed Ch 1)
# 6. (61.564) = 1 and (64.564) = 0.2 so the answer is (0.2) (1)
2.5
2.5
0.58 0.16 = 0.42.
# 7. 2180 grams, 2.64 SDs below average. About .4% of babies born to
nonsmokers are below this weight.
# 8. Le
UC B ERKELEY, S TAT 135
Homework 2 Solution
July 9, 2014
1 Ex. 7. Ch. 2
x=
se(x) =
67
91
x(1 x)
N n
= 0.039
N
n 1
C I = [x 1.96 se(x), x + 1.96 se(x)] = [0.660, 0.813]
2 Ex. 10. Ch. 2
2.1 a.
By CLT, for large n, the probability distribution of
Z=
X
/
Stat135 HW4 Guoling Liu
Ch8 43.
Part A
According to the histogram, the gamma distribution would be a plausible model.
Part B
Moment Method
lambda.mom=0.01266466
alpha.mom=1.012352
Maximum Likelihood Method
alpha.mle=1.012352
lambda.mle=0.01266477
To compa
UC Berkeley, Stat 135
Homework 9 Solution
1. Ex 36. Ch 9
2 = 47.4
df = 11
p < .005
The suicide rate is not constant. There is a seasonal pattern: it is low
in December-March, high in April-June, and low in July.
2. Ex 38. Ch 9
2 = 42.45
m
2 = 50.52
f
df =
Midterm One
Statistics 153, Spring 2014
15 May, 2014
1. Two quarterly time series data sets are plotted below. There are 36 observations (quarterly data for 9
years) in each data set. It is believed that for one of the two data sets, the model:
Xt = (a +
Stat 135 Fall 2014: HW 7 Solutions
1
Let X1 , X2 , . . . , X10 be a random sample of size 10 from a Poisson distribution with mean
. Show that the critical region dened by cfw_ 10 Xi 3 denes a best test for testing
1
H0 : = 0.1 against H1 : = 0.5
Determin
Stat 135 - Homework 10
Yannik Pitcan
December 15, 2014
12.3
We want to nd that minimizes
n
( y i )2 .
i =1
We note that
n
n
i =1
i =1
(yi )2 = cfw_(yi y)2 + (y )2
and this quantity is minimized when = y.
12.10
This follows immediately from the fact t
Stat 135 - Homework 6
Yannik Pitcan
November 10, 2014
9.12
Our likelihood ratio is
n
0 e 0 n X
sup n en X
n
= 0 e0 n x /(1/ xe)n = (0 xe)n e0 n x = (0 e x e0 x )n .
And 0 e is a known quantity, so our rejection rejection will be of the form cfw_ X exp[0
Name: Student ID Number:
Statistics 135 Fall 2007 Midterm Exam
Ignore the nite population correction in all relevant problems. The exam is closed book, but some possibly useful facts about probability distributions are listed on the last page. Show your w
Stat 155 Fall 2014: Homework 1
1. Lost Springs population:
(a) The sample mean takes the values 52, 67, 72.5, and 59.5 (all in years), with an equal probability of 1/4.
(b) The sample mean is not unbiased: E(X) = 62.75, = 62.25
2. (7.4) The rst, since Var
UC Berkeley, Stat 135
Homework 5 Solution
1. Rice Ex. 24. Ch. 9
(a) Generalized Likelihood Ratio is
(x) =
1 x
1 nx
1 2
2
maxp n px (1 p)nx
x
n
x
=
1
2n
x x
n
1
x nx
n
(b) Reject (x) c is equivalent to
x
x
log (x) = n log 2 x log
(n x) log 1
log c
n
n
T
UC Berkeley, Stat 135
Homework 6 Solution
1. Rice Ex. 8. Ch. 11
(a) Is it appropriate to use two-sample t-test with n = m 20?
No, it is not appropriate. The samples of ten readings each within
each group are not independent. So the Xi do not form a random
Stat 135 Fall 2015: Solutions to Homework 5
Assigned problems:
Chapter 8: 18abc, 47abc, 51, 52abc, 58abcde
18a. We cant use E(X) to compute the method of moments estimate of since E(X) is a constant. Therefore,
going to the second moment 2 , we have:
2
1
Stat 135 Fall 2015: Solutions to Homework 1
Assigned problems:
Stat Labs, chapter 1: Numerical, graphical summaries, assessment. Chapter 7: 2, 10, 24
1 Problem from Stat Labs, chapter 1:
Numerical summaries for the baby birthweights(in ounces):
Mother smo
Chapter 10
1 Consider the situation where the average response Y for conditions A and B
are as follows:
Not A
10
12
Average Y
Not B
B
A
20
22
Under this situation, for any given state of condition A (A or Not A), B is
associated with an increase in the av
MLEstimation
STAT 135: Intro to statistical inference
Noureddine El Karoui
Department of Statistics
UC, Berkeley
September 11, 2016
Noureddine El Karoui
STAT 135: Intro to statistical inference
MLEstimation
Introduction to statistical inference
Noureddine
STAT 135: Intro to statistical inference
Noureddine El Karoui
Department of Statistics
UC, Berkeley
August 29, 2016
Noureddine El Karoui
STAT 135: Intro to statistical inference
Introduction to statistical inference
Noureddine El Karoui
STAT 135: Intro to
Some Probability reminders
Stat 135, Fa16
Noureddine El Karoui
Two main tools of probability needed in 135:
Law of large numbers
Central limit theorem
We will also need some manipulation routines concerning expectation and variance
1
Expectation
For a d
256
Chapter 8
Estimation of Parameters and Fitting of Probability Distributions
8.3 Parameter Estimation
Poisson distribution as a model for random counts in space or time rests on three
assumptions: (1) the underlying rate at which the events occur is co