PRE REQUISITE CONCEPTS
Expectation
Discrete and
Continuous
random
Variables
If X is discrete random variable with frequency function
p(x)(probability mass function) the expected value of X
denoted by E(X) is
provided
.
If the sum diverges the expectation

Stat 135 Fall 2016
SECTION QUIZ
Section time: 12-2pm
Name:
September 12, 2016
SID:
In the following, make sure to explain your reasoning, or points will be taken off. Problems 1-4 are 1 point each, and
the others are 3 points each, for a total of 10 point

Solution for Quiz 1 Section 1 (12:00-2:00pm)
STAT 135 Fall 2016
September 26, 2016
Problem 1
True. Because X1 and X100 are drawn independently from the same population.
Problem 2
False. Sample mean is a random variable but the variance of it is a constant

Name:
ID:
Homework for 1/27 Due 2/5
1. [8-13] In Example D of Section 8.4, the pdf of the population distribution
is
1 + x 1 x 1
2
f (x|) =
,
1 1,
0
otherwise
and the method of moments estimate was found to be
= 3X (where X
is the sample mean of the ra

Stat 135 Fall 2016
SECTION QUIZ
Section time: 12-2pm
Name:
Score:
October 3, 2016
SID:
/10
1. Suppose that X is a discrete random variable with P (X = 1) = and P (X = 0) = 1 . Four independent observations
of X are made, with x1 = 1, x2 = 0, x3 = 1, x4 =

Stat 135 Fall 2016
SECTION QUIZ
Section time: 2-4pm
Name:
Score:
October 3, 2016
SID:
/10
1. Suppose that X is a discrete random variable with P (X = 1) = and P (X = 2) = 1 . Four independent observations
of X are made, with x1 = 1, x2 = 1, x3 = 2, x4 = 2

Solution for Quiz 1 Section 2 (2:00-4:00pm)
STAT 135 Fall 2016
September 26, 2016
Problem 1
False. Because by sampling without replacement, the element chosen by X1 will not be chosen by
X100 , then the distribution of X100 depends on X1 .
Problem 2
True.

Solution for Quiz 3 Section 1 (12:00-2:00pm)
STAT 135 Fall 2016
October 6, 2016
Problem 1
= 3/4.
(a) We have E[X] = . Then the method of moments estimate is = X
(b) The likelihood function is lik() = 3 (1 ). The log likelihood function is
l() = 3 log() +

STAT 135 Fall 2016
Solutions for Homework 4
Table 1: Comparison between observed frequency and expected frequency
Number of Hops xi
1
2
3
4
5
6
7
8
9
10
11
12
Frequency fi
48
31
20
9
6
5
4
2
1
1
2
1
Expected

Stat 135 Fall 2016
SECTION QUIZ
Section time: 12-2pm
Name:
Score:
September 19, 2016
SID:
/10
In the following, make sure to explain your reasoning, or points will be taken off.
1. The center of a 95% confidence interval for the population mean is a rando

Solution to Homework 3
STAT 135 Fall 2016
Dong Yin
October 11, 2016
7.26
(a)
n
N
j=1
i=1
X
1X
= 1
X
Xj =
Ui xi ,
n
n
since for each i, Ui xi = 0 if there exists no Xj such that Xj such that Xj = xi , otherwise
Ui xi = xi = Xj for some j.
(b)
P (Ui = 1) =

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Interval Estimates for population parameters
So far, we have seen confidence intervals for the population mean using the sample mean as an estimator.
The central limit theorem tells us that for \(n\) and \(

Solution for Quiz 2 Section 1 (12:00-2:00pm)
STAT 135 Fall 2016
October 6, 2016
Problem 1
True. Confidence interval is determined by the sample, which are random. So confidence interval
is also random.
Problem 2
False. The 95% confidence interval contains

Stat 135 Fall 2016
SECTION QUIZ
Section time: 2-4pm
Name:
Score:
September 19, 2016
SID:
/10
In the following, make sure to explain your reasoning, or points will be taken off.
1. A 95% confidence interval for contains the sample mean with probability 0.9

Solution for Quiz 3 Section 2 (2:00-4:00pm)
STAT 135 Fall 2016
October 6, 2016
Problem 1
= 1/2.
(a) We have E[X] = +2(1) = 2. Then the method of moments estimate is = 2X
(b) We have the likelihood function
lik() = 2 (1 )2 .
The log likelihood function is

Solution for Quiz 2 Section 2 (2:00-4:00pm)
STAT 135 Fall 2016
October 6, 2016
Problem 1
False. By the method of constructing confidence intervals in the textbook, the confidence interval
have center at the sample mean. So it always contains the sample me

STAT 135 Solutions to Homework 4: 30 points
Spring 2015
Problem 1: 10 points
In each of the following cases, (i) write down the likelihood function
of, (ii) show that the corresponding T (X)
is a sufficient statistic, (iii) compute M LE and (iv) compute E

Stat 135 Fall 2016
SECTION QUIZ
Section time: 2-4pm
Name:
September 12, 2016
SID:
In the following, make sure to explain your reasoning, or points will be taken off. Problems 1-4 are 1 point each, and
the others are 3 points each, for a total of 10 points