6.1
Chapter 6
Point Estimation
1
6.1
6.1 Some General Concepts of Point Estimation
A point estimate of a parameter is a single number
that can be regarded as a sensible value for . A point
estimate is obtained by selecting a suitable statistic
and computi
FORMULA SHEET
CHAPTER 2
Law of Total Probability: P( B) PB | A1 P A1 . PB | An P An ;Bayes: P Ai | B
Conditional Probability: P A | B
PB | Ai P Ai
P( B)
P A B
; Union of 2 events: P A B P( A) P( B) P A B ;
P( B)
Union of 3 events: P A B C P( A) P( B)
9.1
Chapter 9
Inferences Based on Two
Samples
1
9.1
9.2 z Tests and Confidence Intervals for a
Difference Between Two Population Means
In many applications, it is of interest to study the
difference between the means of two populations. In
fact, by studyi
Note: all denitions are from your textbook
CHAPTER 1
What is Statistics?
The discipline of statistics teaches us how to make intelligent judgments and informed decision in the presence of uncertainty and variation.
Branches of Statistics
Descriptive Stat
Chapter 1: Overview and Descriptive Statistics
CHAPTER 1
Section 1.1
1.
a.
Houston Chronicle, Des Moines Register, Chicago Tribune, Washington Post
b.
Capital One, Campbell Soup, Merrill Lynch, Pulitzer
c.
Bill Jasper, Kay Reinke, Helen Ford, David Menede
3502 Tutorial 3
1. Compute the following binomial directly from the binomial formula:
a. b(3; 8, .35) b. b(5; 8, .6), c. P (3 X 5) when X Bin(7, .6).
2. A company that produces ne crystal knows from experience that 10% of its goblets have
cosmetic aws and
Chapter 3
Discrete Random Variables
and Probability Distributions
1
3.3 Expected Values
Definition
Let X be a discrete rv with set of possible values D
and pmf p (x). The expected value or mean value of
X, denoted by E(X) or X or just , is
E(X) = X =
2
Ex
9.1-9.2
Chapter 9
Inferences Based on Two
Samples
1
9.1-9.2
9.2 z Tests and Confidence Intervals for a
Difference Between Two Population Means
In many applications, it is of interest to study the
difference between the means of two populations. In
fact, b
1. When human beings assemble widgets, the proportion of defective widgets is known to be
0.04. A robot assembled 500 widgets and 12 of them were defective. Is there evidence to
support the hypothesis that robots are better at assembling widgets? If so, a
1. H0: p =0.04, Ha : p < 0.04, z =1.83.
2.
A. E(
)= k/k =
B. E ([n1
/[n1+n2+ .+nk])= [n1+.+ nk]/[ n1+n2+ .+nk] =
3.
= 1, not in terms of .
A.
B.
=
4.
A. k = 6/7
B. fX(x) = 2/7 [(x + 1)3 x3] , fY(y) = 2/7 [(y + 1)3 y3]
C. E(X) = E(Y) = 9/14, 2X = 2Y = 19
Review
C. Gravel (Carleton)
STAT 3502
January 2015
1/1
Review
C. Gravel (Carleton)
STAT 3502
January 2015
2/1
Review
C. Gravel (Carleton)
STAT 3502
January 2015
3/1
Review
C. Gravel (Carleton)
STAT 3502
January 2015
4/1
Review
C. Gravel (Carleton)
STAT 35
STAT 3502
Introduction to Probability and Statistics for
Engineers
Instructor: Christopher Gravel
Carleton University
STAT 3502
C. Gravel (Carleton)
STAT 3502
January 2015
1 / 26
Introduction
Introduction
Statistics
Statistics is the science of learning f
8.1-8.4
Chapter 8
Tests of Hypothesis Based on a
Single Sample
1
8.1-8.4
8.1 Hypotheses and Test Procedures
A statistical hypothesis, or just hypothesis, is a
claim or assertion either about the value of a single
parameter (population characteristic or ch
5.1
Chapter 5
Joint Probability Distributions
and random Samples
1
5.1
5.1 Jointly Distributed Random Variables
Many studies/applications involve multiple random
variables. In this section, we consider joint
probability distributions for two discrete rv's
5.2-5.5
Chapter 5
Joint Probability Distributions
and random Samples
1
5.2-5.5
5.2 Expected Values, Covariance, and
Correlation
Let X and Y be jointly distributed rvs with pmf
p(x, y) or pdf f (x, y) according to whether the
variables are discrete or cont
Tutorial 7. Sec. 5.1-5.2
1. A service station has both self-service and full-service islands. Let X be the number of
hoses being used on the self-service and Y the number of hoses on the full-service on a
particular time of the day. The joint pmf of (X, Y
Tutorial 6 Solutions
1. Read from the normal table. We get respectively a) .9842, b) .9842-.5=.48.
2. We get respectively a) 2.14, b) 1.17.
3. a) P (X 15) = .5.
b)
P (|X 15| 3) = P (|Z| 3/1.25) = P (|Z| 2.4) = .9918 .008 = .98
4. We can use normal approxi
Solutions to Review of Ch. 6-8
1. The likelihood function is
L = L(X1 , . . . , Xn ; , ) = f (X1 ; , ) f (Xn ; , )
n exp ( n Xi ) en , if min(Xi ) ,
i=1
=
0,
else
The loglikelihood function is
= (X1 , . . . , Xn ; , ) = ln(L) =
n ln()
n
i=1
if min(Xi ) ,
Review of Ch. 6-8
1. Consider a random sample X1 , . . . , Xn from shifted exponential
e(x) , x
f (x; , ) =
0,
else.
Obtain the MLE and . Check if MLE of is biased.
2. High concentration of the toxic element arsenic is all too common in groundwater. An
a
Carleton University
School of Mathematics and Statistics
STAT 3502 A Probability and statistics Summer 2015
Instructor: Ahmed Almaskut
Office: 5218 Herzberg Laboratories
Tel.: (613) 520-2600 ext. 8999
E-mail: aalmasku@math.carleton.ca
Lectures: Mondays an
Chapter 4
Continuous Random Variables
and Probability Distributions
1
4.1 Probability Density Functions
Recall from Chapter 3 that a random variable X is
continuous if
possible values comprise either a single interval
on the number line (for some A < B,
7.1-7.4
Chapter 7
Statistical Intervals Based on a
Single Sample
1
7.1-7.4
7.1 Basic Properties of Confidence Intervals
Suppose that of interest is the population mean, , and
that the following is true:
1. The population understudy is normal.
2. The value
Chapter 2
Probability
2.1 Sample Spaces and Events
Definition: An experiment is any activity or process
whose outcome is subject to uncertainty.
Examples:
Tossing a coin and observing the outcome.
Tossing a six-sided die.
Measure of the time it takes t
6.1-6.2
Chapter 6
Point Estimation
1
6.1-6.2
6.1 Some General Concepts of Point Estimation
A point estimate of a parameter is a single number
that can be regarded as a sensible value for . A point
estimate is obtained by selecting a suitable statistic
and
Announcements
The Paul Menton Center is still looking for a volunteer note taker
to assist students in this class.
If interested in helping contact
volunteer_notetaking@carleton.ca or go to 501 UC
TA ofce hours
Yanjiang Yu SA 316 Friday 12:30-1:30 pm
Dani
Ch. 5 Jointly Distributed Random Variables and Random Samples
5.3 Statistics and Their Distributions
5.3 Statistics and Their Distributions
Thus far, we have focused on the underlying theory and
mathematics needed to describe the behaviour of random
pheno
Ch. 4 Continuous Random Variables and Probability Distributions
The Normal Distribution - 6
C. Gravel (Carleton)
STAT 3502
January 2015
1 / 32
Ch. 4 Continuous Random Variables and Probability Distributions
Last Class
We considered the Normal distribution
STAT 3502
Assignment # 2
Total mark=30
Due: 18 July, 2012 in tutorial
1. The weekly demand for propane gas (in 1000s of gallons) from a particular facility is
a r.v. x with pdf
(
2(1 x12 ) 1 x 2
f (x) =
0
otherwise.
a. [1] Compute the cdf of x.
b. [1] Wha
3502 Assignment 1 due Wed. Feb. 6th in class. No late assignment will be
accepted. TAs will NOT accept assignments directly from students.
1. An airliner has 2 p.m., 8 p.m., and 10 a.m. flights from Montreal to Frankfort, Paris, and
Vancouver respectively
El c.
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STAT 3502
Tutorial # 7
Winter 2012
1. On the basis of extensive tests, the yield point of particular type of mild steel-reinforcing
bar known to be normally distributed with = 100. The comparison of the bar has
been slightly modified, but the modification
STAT3502
Tutorial # 3
1. A system consists of two identical pumps, #1 and #2. If one pump fails, the system
will still operate. However, because of the added strain, the extra remaining pump is
now more likely to fail than was originally the case. This is