ST221:
Introduction to Probability
Chapter 2 - Axioms of Probability
Seung Jun Shin
Department of Statistics
Korea University
Sample Space and Events
Chapter 2
Sample space is the set of all possible outcomes of an
experiment and denoted by S.
- Sex of a
STAT221: INTRODUCTION TO PROBABILITY THEORY
Solutions to Homework 3
1. The daily demand, D is Binomial (10, 1/3). Let Y denote the number of papers
sold during the day, and b denote the number of papers he purchased.
Consider possible bs.
(i) When b=2,
y
ST221:
Introduction to Probability
Chapter 6 - Jointly Distributed RV (part II)
Seung Jun Shin
Department of Statistics
Korea University
Sum of Indep RVs
Chapter 6
Suppose X and Y are independent continuous RVs having
PDF fX and fY . Then
FX +Y (a) = P(X
ST221:
Introduction to Probability
Chapter 4 - Random Variables (part I)
Seung Jun Shin
Department of Statistics
Korea University
Random Variables
Chapter 4
We are often interested in some function of outcome.
ex) When tossing two dice, we may be interest
ST221:
Introduction to Probability
Chapter 5 - Continuous Random Variables (part II)
Seung Jun Shin
Department of Statistics
Korea University
Exponential Random Variables
Chapter 5
An exponential random variable X has the PDF f (x )
defined as
f (x ) = e
ST221:
Introduction to Probability
Chapter 3 - Conditional Probability and
Independence (part II)
Seung Jun Shin
Department of Statistics
Korea University
Independent Events
Chapter 3
In general, the conditional probability P(E |F ) is
different from the
ST221:
Introduction to Probability
Chapter 5 - Continuous Random Variables (part I)
Seung Jun Shin
Department of Statistics
Korea University
Continuous Random Variable
Chapter 5
A random variable X is said to be continuous or have
a continuous distributio
ST221:
Introduction to Probability
Chapter 3 - Conditional Probability and
Independence (part I)
Seung Jun Shin
Department of Statistics
Korea University
Introduction
Chapter 3
Conditional probability is one of the most important
concept in probability an
STAT221: INTRODUCTION TO PROBABILITY THEORY
Solutions to Homework 5
1. For any nonnegative integers m and n,
P( X = m, Y = n) = (1 p)m p(1 p)n p = (1 p)m+n p2 .
Thus, the joint mass function of X and Y is
(
p( x, y) =
2.
(1 p ) x + y p2
for x, y = 0, 1,
ST221:
Introduction to Probability
Chapter 7 - Expectation and Its Properties
Seung Jun Shin
Department of Statistics
Korea University
Expectation
Chapter 7
Recall that
E (X ) =
X
xp(x ),
for discrete X
x
Z
=
xf (x )dx ,
for continuous X
x
If X and Y have
ST221:
Introduction to Probability
Chapter 6 - Jointly Distributed RV (part I)
Seung Jun Shin
Department of Statistics
Korea University
Joint Distribution Functions
Chapter 6
The joint cumulative probability distribution function
of two random variables X
ST221 - Homework 4
1. The PDF of X, the lifetime of a certain type of electronic device (measured in hours), is given by
f (x) = 10/x2 ,
x > 10
(a) Find P (X > 20).
(b) What is the CDF of X?
(c) What is the probability that of 6 such types of devices, at
ST221:
Introduction to Probability
Chapter 1 - Combinatorial Analysis
Seung Jun Shin
Department of Statistics
Korea University
Introduction
Chpater 1
Counting plays an essential role in computing
probability.
Mathematical theory of counting is formally kn
ST221 - Homework 3
1. A newsboy purchases papers at 10 cents and sells them at 15 cents. However, he is not allowed to return
unsold papers. If his daily demand is a binomial random variable with n = 10 and p = 1/3, approximately
how many papers should he
ST221 - Homework 6
1. The time that it takes to service a car is an exponential random variable with rate 1.
(a) If A.J. brings his car in at time 0 and M.J. brings her car in at time t, what is the probability that M.J.s
care is ready before A.J.s care?
STAT221: INTRODUCTION TO PROBABILITY THEORY
Solutions to Homework 4
1.
(a)
(b)
(c)
R
10
= 1/2
dx = x10 ]20
20 x2
R y 10
F (y) = 10 x2 dx = 1 10
y, y >
6 i
i
6
= 1 73/36
6i=3 ( i )( 23 ) ( 13 )
10
since F (15) = 15 .
10. F (y) = o f or y < 10.
= 1 0.10013
ST221 - Homework 1
1. Give an analytic proof of
n
n1
n1
=
+
,
r
r1
r
1 r n.
2. Provide a combinatorial argument to prove
X
r
n+m
n
m
=
.
r
i
ri
i=0
Hint: Consider a group of n men and m women. How many groups of size r are possible?
3. Using the bin
ST221:
Introduction to Probability
Chapter 4 - Random Variables (part II)
Seung Jun Shin
Department of Statistics
Korea University
Poisson Random Variable
Chapter 4
A random variable X cfw_0, 1, 2, is said to be a
Poisson Random Variable with parameter >
STAT221: INTRODUCTION TO PROBABILITY THEORY
Solutions to Homework 6
1.
1 t
e .
2
(b) 1 3e2 .
(a)
2. Let X denote Jills score and let Y be Jacks score. Also, let Z denote a standard
normal random variable.
(a)
P (Y > X ) = P (Y X > 0 )
= P(Y X > 0.5)
"
#
Y
ST221 - Homework 2
1. Suppose that an insurance company classifies people into one of three classes: good risks, average risks, and
bad risks. The companys records indicate that the probabilities that good-, average-, and bad-risk persons
will be involved
STAT221: INTRODUCTION TO PROBABILITY THEORY
Solutions to Homework 2
1. Choose a person at random
Pcfw_they have accident = Pcfw_ acc. | good Pcfw_ g + Pcfw_ acc. | ave. Pcfw_ ave.
+ Pcfw_ acc. | badP(b)
= (.05)(.2) + (.15)(.5) + (.30)(.3) = .175
.95(2)
Pc
ST221:
Introduction to Probability
Chapter 6 - Jointly Distributed RV (part III)
Seung Jun Shin
Department of Statistics
Korea University
Distribution of Function of RVs
Chapter 6
Let X1 and X2 be continuous RVs with a joint density
fX1 ,X2 .
For a one-to
ST221 - Homework 5
1. Consider a sequence of independent Bernoulli trials with success probability p. Let X1 be the number of
failures preceding the first success, and let X2 be the number of failures between the first two successes. Find
the joint probab
STAT221: INTRODUCTION TO PROBABILITY THEORY
Solutions to Homework 1
1.
n1
r
!
n1
r1
+
!
( n 1) !
( n 1) !
+
r!(n 1 r )! (n r )!(r 1)!
!
n!
nr
r
n
=
+
=
r!(n r )!
n
n
r
=
2.
n+m
r
There are
!
groups of size r, since there are
n
i
!
of size r that consist o
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Learning Objectives
When you have completed
this chapter, you will be
able to:
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Multiple Regression
Analysis
LO1 Describe the relationship
between several independent
variables and a dependent
variable
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Correlation and
Linear Regression
13
Learning Objectives
When you have completed
this chapter, you will be
able to:
LO1 Define the terms
independent variable and
dependent variable.
LO2 Calculate, test,
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Learning Objectives
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Page 410
Analysis of Variance
When you have completed
this chapter, you will be
able to:
LO1 List the characteristics of
the F distribution and locate
values in an F table.
LO2 Perform a te
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Two-Sample Tests
of Hypothesis
11
Learning Objectives
When you have completed
this chapter, you will be
able to:
LO1 Test a hypothesis that
two independent population
means with known population
standard
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Learning Objectives
When you have completed
this chapter, you will be
able to:
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Time Series
and Forecasting
LO1 Define the components
of a time series.
LO2 Compute a moving
average.
LO3 Determine a li
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Learning Objectives
When you have completed
this chapter, you will be
able to:
LO1 Conduct a test of
hypothesis comparing an
observed set of frequencies to
an expected distribution.
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Nonparametric
Met