Stats 120B HW 2: Solutions
1. [15 pts] Let I() be the indicator function. Prove IAB = IA IB for any two sets
A and B.
For any x A B, IAB (x) = 1 by definition. Note that x A B implies X A
and X B, so IA (x) = IB (x) = 1. So the results holds for x AB. Now
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, 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
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
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
=
() =
( )
( )
=
(
W dtzwkzma
Problem 1:
Consider two
random variables, X ~ Beta(2017, 2017), and Y ~ Poisson(1).
Here are some facts that might be useful'
The pdf of Beta(a ) is fX(.'B) _ $91
'_  , (1 l B , . T ' i
P (Y k) Ake A/kL Theng of Poisson(A)is M/y(1)a=;xp(,\(
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
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
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: Final Exam, Winter 2016
Version A
Homework ID#:
Name:
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 except for four 8
Stats 120B: Midterm Exam, Winter 2017
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 except for one
Stats120BHW37
plot_sample_means < function(f_sample, n, m=300,title="Histogram", .) cfw_
# define a vector to hold our sample means
means < double(0)
# generate 300 samples of size n and store their means
for(i in 1:m) means[i] = mean(f_sample(n,.)
#
1. Assume X1 , . . . , Xn form a random sample from N (, 2 ) with both and 2
unknown.
State the result. You dont have to
(a) What is the sampling distribution of X?
prove it.
N (, 2 /n)
X
(b) State a result P
about the distribution of an expression invo
Stats 120B Homework 5: Solutions
1. Let X1 , . . . , Xn be a random sample from Geometric distribution with pmf
P (X = x) = (1 p)x p.
The mean of this distribution is (1 p)/p.
(a) Find the estimator for p using method of moments.
n , we get
Match (1 p)/p
Stats 120B HW 4: Solutions
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?
We have a random sample X1 , . . . , X16 following discrete unif
Stats 120B HW 3: Solutions
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 ) will
Stats 120B Homework 1: Solution
1. Let X and Y be two independent random variables following beta distributions
Beta(120, 2017).
(a) [5 pts] Whats P(X = 0.5)?
Continuous distribution, 0.
(b) [5 pts] Whats P(X + 3 < 2Y )?
2Y is between 0 and 2, X + 3 is al
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
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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