STAT 3502 (Winter Term 2013)
Solutions to Assignment 1
1. (a) P (A B C) = P (A)P (B)P (C) = (.7)(.6)(.8) = .336.
(b) P (at least one flight is full) = 1P (no flight is full) = 1P (A0 )P (B 0 )P (C 0 )
= 1 (.3)(.4)(.2) = 1 .024 = .976.
(c) P (A0 B 0 C) = P
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
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)
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
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
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
STAT 3502
Tutorial # 4
1. A mail-order computer business has six telephone lines. Let X denote the number of
lines in use at a specic time. Suppose the pmf of X is as given in the accompanying
table
x
0
p(x) .10
1
.15
2
.20
3
.25
4
.20
5
.06
6
.04
Calcula
STAT 3502
Tutorial # 6
1. A college professor never nishes his lecture before the end of the hour and always
nishes his lectures within 2 min. after the hour. Let X= the time that elapses
between the end of the hour and the end of the lecture and suppose
STAT 3502
Tutorial # 7
1. Suppose the time spend by a randomly selected student who uses a terminal connected
to a local time sharing computer facility has a gamma distribution with mean 20 min
and variance 80 min2 ?
a. What are the values of and ?
b. Wha
STAT 3502
Tutorial # 8
1. Let X1 , X2 , , Xn represent a random sample from a Rayleigh distribution with pdf
f (x; ) =
x x2 /(2)
e
x>0
a. It can be shown that E(X 2 ) = 2. Use this fact to construct an unbiased estimator
of based on
Xi2 .
b. Estimate from
STAT 3502
Tutorial # 9
1. In an experiment designed to measure the time necessary for an inspectors eyes to
become used to the reduced amount of light necessary for penetrant inspection, the
sample average time for n = 9 inspectors was 6.32 sec and the sa
STAT 3502
Tutorial # 6-Summer 2011
1. Suppose the time spend by a randomly selected student who uses a terminal connected
to a local time sharing computer facility has a gamma distribution with mean 20 min
and variance 80 min2 ?
a. What are the values of
1. We have the following probabilities for two events A1, A2 such that P (A1) = 0.12, P(A2) 2'
0.30, P(A1 m A2) = 0.07. Find 13(A1 m Ag), P(A n A)
M
_,_.W.w.~m~
"D A/(/ F) (/U 5 I'M [3(IUAZ)
)( 1) al1wL7>(31>)1~)(/ilg)~/? WA)
2. A box contains 5 defecti
Practice Question - 2
1. An operator receives on average 2 emergency calls every 3 minutes. What is the probability of receiving
5 calls or more in a 9 minute period?
9 1
2i: $156.11? EMQVS'Q- (116? m 5/ mm Jm'jml ()
PKX)1)lw-I(K4) I rilelx nglgzgj; (111
Formula-Test 2
E(X) = =
P
x
2 =
xp(x),
P
x (x
X,Y =
V (a1 X1 + a2 X2 + + an Xn ) =
Pn Pn
Gamma f (x) =
1
x1 ex/ ,
()
P
x
x2 p(x) 2
Cov(X,Y )
X Y
Cov(X, Y ) = E(XY ) X Y ,
i=1
)2 p(x) =
j=1
x>0
ai aj Cov(Xi , Yj )
E(X) = , V (X) = 2
Exponential f (x) =
Selected Formulae for Test 1
= E(X) =
X
Z
xp(x) (X discrete),
=
xf (x)dx (X continuous)
x2 = V (X) = E[(x )2 ] = E(X 2 ) (E(X)2
If A1 , A2 , , Ak be mutually exclusive and exhaustive events. Then for an event B,
P (B) = P (B|A1 )P (A1 ) + + P (B|Ak )P (A
Tutorial # 2
1. An individual is presented with three dierent glasses of cola, labeled C, D, and P. He
is asked to taste all three and then list them in order of preference. Suppose the same
cola has actually been put into all three glasses.
a. What are t
Solution Review problems
1. X N (500, 1002 )
a) P (X > 605) = P (Z > 1.05) = 1 (1.05) = 0.1467
500
x0 = 544
b) P (X < x0 ) = 0.67 P (Z < z0 ) = 0.67 z0 = 0.44 0.44 = x0100
605500
2
c) n = 9 X N (500, 100 /9) then P (X > 605) = P (Z > 100/3 = 1 (3.15) =
1
Review problems
1. The scores on an admission test the Engineering Program of a certain university are
normally distributed with a mean of 500 and a standard deviation of 100.
a) What is the probability that a randomly selected student will obtain a score
STAT 3502
Tutorial # 5
1. An instructor who taught two sections of engineering statistics last term, the rst with
20 students and the second with 30, decided to assign a term project. After all projects
had been turned in, the instructor randomly ordered
CARLETON UNIVERSITY
August 2014
Final
EXAMINATION
August 2014
DURATION:
3
No.
HOURS
Department Name & Course Number:
Course Instructor(s)
Dr.
of Students
Mathematics and Statistics STAT3502A Final
Z. Montazeri
AUTHORIZED MEMORANDA
NON-PROGRAMMABLE CALCULA
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 3
Random Variable
A rule that associate a number to each outcome of an experiment (or each outcome in S) is
random variable.
Bernoulli random variable: Any random variable whose only possible values are 0 and 1
Example: Give three examples of Bern
CHAPTER 2
PROBABILITY
Probability is used in inference statistics as a tool to make statement for population from
sample information.
Experiment is a process for generating observations
Sample space is all possible outcomes of an experiment.
Event is a
Chapter 3 -Discrete Random Variables and Probability Distributions
3.3 Expected Values
3.3 Expected Values
Denition: The Expected Value of X p. 107
Let X be a discrete rv with set of possible values D and pmf p().
The expected value or mean value of X, de
STAT 3502 (Winter 2015)
SOLUTIONS TO ASSIGNMENT 2
1. (a) The event of selling x models with side airbags and 4 x models without side airbags can
4
occur in x ways, where x can be 0, 1, 2, 3 or 4. Therefore
p(x) = P(X = x) =
4 1
,
x 24
for x = 0, 1, 2, 3,
Stat 3502
Assignment 2
Due In Tutorial
Carleton University
School of Mathematics and Statistics
STAT 3502: Probability and Statistics - Assignment 2
Sections A due Wednesday, Feb 26, 2014 In Tutorial
Section B due Tuesday, Feb 25, 2014 In Tutorial
INSTRUC
1
Formulae Sheet
Pn 2 (Pni=1 xi )2
P
n
X
x
1
xi
n
x =
, s2 =
(xi x)2 = i=1 i
n
n 1 i=1
n1
P
P
P
E(X) = = x xp(x),
2 = x (x )2 p(x) = x x2 p(x) 2
If A1 , A2 , , Ak be mutually exclusive and exhaustive events. Then for an event B,
P (B) = P (B|A1 )P (A1 )
STAT 3502A
Practice Questions 2
Summer 2017
What is the probability that an exponential random variable falls between two standard deviations of its mean?
(a) 0.0498
(b) 0.9502 (*)
(c) 0.905
(d) 0.75
Suppose X has probability density function
(
x1 , 0 x
Chapter 6
The Normal Distribution
Continuous Random Variables
As mentioned earlier, a continuous random
variable is a random variable that can take on any
value in some continuous interval(s) of values.
Continuous random variables are used when the
event
CHAPTER 4
Continuous Random Variables and Probability Distributions
z Basic definitions and properties of continuous random variables
Continuous random variable: A random variable is continuous if its set of possible values
is an entire interval of number
Chapter 4 (4.1 4.7)
Probability
Outline
These notes will cover the following concepts:
Sample Spaces and Events
Rules for Probability
Probability Trees
Counting Rules
Conditional Probability and Independence
Bayes Rule
Experiments
An experiment is a