Stats 120B Homework 3: Due Thursday, Jan. 28
1. Done in lecture 7.
2. Let cfw_Xn be a sequence of random variables with CDF dened by
n=1
FXn (x) = 1
1
, 1 x < .
xn
(a) [5 pts] What is the limiting distribution of Xn ?
limn FXn (x) = limn 1 x1 = 1
n
lim
Homework 3 (Solution)
1
1
1. Let be the roundoff error of the th number. Then ~ [ 2 , 2], with
0.5
0.5
2
= [ ] = = 
=0
2 0.5
0.5
and
0.5
0.5
3
1
= [ ] = ( 0) = 
=
3 0.5 12
0.5
2
2
If 100 numbers are added, then let = 100 denote the roundoff error, t
Homework 1 (Solution)
1. has a Gamma distribution with parameter and , so let () denote the pdf for , we have
() =
1
()
By class notes (Thm 3.8.2), if = with > 0, let () denote the pdf for , then () =
1
( ), so
1 1
1 1 ( ) 1
=
() =
( )
( )
=
(
Ch 4 Describing Numerical Data (p 49)

Objectives:
o activities that require an understandment of variation
o methods used 2 display & summarize num. data
methods characterize:
whats typical about data (average)
how data varies around typical value (a
120B: Jan 23 and 25, 2017
HW1
Finishing Bivariate normal distribution (Ch 5.10)
Review: independence, correlation, CDF.
Largesample theory (Ch 6)
Textbook too hard to read?
Ref:
Rice, J. (2006). Mathematical Statistics and Data Analysis
Classic: Statisti
Stats 120B Homework 4: Due Mon, Feb. 13
1. Suppose that 16 digits are chosen at random with replacement from the set cfw_0, . . . , 9.
What is the probability that their average will lie between 4 and 6?
2. Suppose that X1 , . . . , Xn form a random sampl
Topics for this week
Indicator function
Continuous vs. discrete distribution
Another example of function of r.v.
Revisit Normal distribution (Ch 5.6)
Bivariate normal distribution (Ch 5.10)
Indicator function
Indicator function I() takes 1 if True; 0 if F
More on bivariate (multivariate) normal distribution
Let X1 , X2 follow a bivariate normal distribution, and
Y1 = a1 X1 + a2 X2 + a3 and Y2 = b1 X1 + b2 X2 + b3 . Then
Y1 , Y2 follow a bivariate normal distribution.
More generally, let X1 , . . . , Xd fol
120B: Feb 6 and 8, 2017
Midterm
HW3
Estimation (Ch 7)
Midterm
Midterm exam is on Feb 15, in class.
I will send you a seating chart.
No books, 1 page of notes, A4 size.
Review session: Feb 13
Last years exam posted on the web.
HW 4 has some review questio
Stats 120B Homework 3: Due Feb. 1
1. [10 pts] Suppose that the random variables X1 and X2 have a bivariate normal
distribution with means 1 = 2 = 0, variances 12 = 22 = 1 and correlation
(1, 1). Determine the value of the constant b for which Var(X1 bX2
Stats 120B/Math 131B: Midterm Exam, Winter 2016
Version A
Name:
Homework ID#:
UCI Student ID#:
Please read the following instructions carefully:
1. You must show your work/reasoning to receive full credit.
2. The exam is closed book and closed notes excep
120B: Jan 30 and Feb 1, 2017
HW2
Finishing Largesample theory (Ch 6)
Review: independence
Starting estimation (Ch 7)
Homework 2. Q1
Let I() be the indicator function. Prove IAB = IA IB for
any two sets A and B.
Only one step: Discuss all the possibilitie
Stats 120B Homework 2: Due Jan. 25
1. [15 pts] Let I() be the indicator function. Prove IAB = IA IB for any two sets
A and B.
2. Let X and Y follow a bivariate normal distribution with means (3, 2), variances
(1, 4) and covariance c. In other words,
X
3
STATS 120B, W. Shen, Winter 2016
Midterm
Next Tues, 3:40  4:50.
Two frontandback sheets.
Five problems in total. Covers materials till today.
Practice exam/hw/inclass examples.
As I promised, you have seen 95% of the questions. with
changes of d
STATS 120B, W. Shen, Winter 2016
Several common mistakes from the midterm
(1) WLLN, CLT, use !
p
d
Should be using ! and !
(2) Zero covariance implies independence
Need to say bivariate normal distribution
(3) Likelihood. Write
Should write
n
Q
xi
Q
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STATS 120B, W. Shen, Winter 2016
For today
Homework review
More on MLE
1
STATS 120B, W. Shen, Winter 2016
Again, homework is hard. Dont get frustrated.
Discuss with your classmates, come to discussion/office hours.
One common mistake: say, whats the
STATS 120B, W. Shen, Winter 2016
Midterm exam
Midterm exam is on Feb 9, in class.
Start ten minutes late, 3:40  4:50 pm.
I will send you a seating chart.
No books, two pieces of notes, A4 size.
Review session: Feb 4
HW 4 is the practice exam. Wen
STATS 120B, W. Shen, Winter 2016
Midterm next Tuesday
Review session this Thursday
HW4 due Thursday next week.
1
STATS 120B, W. Shen, Winter 2016
Quick review
MLE (most useful; maximize the likelihood)
Method of moments (simple; match the moments)
STATS 120B, W. Shen, Winter 2016
Homework 1.
Question 1(b), X Gamma(, ), find the dist of Y = cX.
(1) Use mgf.
MY (t) = E(ecXt ) = E(eXct ) view ct as a new t
t
.
Note that MX (T ) = 1
ct
t
So MY (t) = 1
= 1
(/c)
Hence Y Gamma(, /c).
1
STATS 120B, W
STATS 120B, W. Shen, Winter 2016
Find your homework based on your ID number.
ALL of the exam questions will come from your
homework/slides/practice exams.
Group discussion for hw is highly encouraged, though you have
to finish it independently.
Pay at
STATS 120B, W. Shen, Winter 2016
Hypothesis testing: what is it about?
Statistical inference: use data (sample) to study/describe the
population characteristics
Estimation: estimate the value of unknown parameters.
Point estimate, interval estimate
Wh
Midterm
You have a chance to redo the midterm problems (excluding
bonus question 1)
Its due on next Tues, Feb 16.
I will take the average of these two scores
If you choose not to do it, I will use your score from inclass
exam then.
I understand the imp
STATS 120B, W. Shen, Winter 2016
HW4 Problem 3
Suppose that X1 , . . . , Xn form a random sample from a
uniform distribution with the following pdf
f (x) =
1
I( x 2)
(f) Whats the mode of f (x)?
Find x that maximizes f .
Any x between and 2 will do.
STATS 120B, W. Shen, Winter 2016
Midterm next Tuesday
Review session this Thursday
HW4 due Thursday next week.
1
STATS 120B, W. Shen, Winter 2016
Quick review
MLE (most useful; maximize the likelihood)
Method of moments (simple; match the moments)
Stats 120B/Math 131B: Final Exam Solutions and Grading
Version A Problem 1/Version B Problem 3:
iid
Assume X1 , . . . , Xn P ois().
(a) [3 pts] State the likelihood function of given observed data x = (x1 , . . . , xn ).
Answer:
n
n
e xi
en i=1 xi
=
n
xi
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