Chapter 6
Ch
6:
Simulation
Homework Solutions 6.3, 6.11 and 6.12
2
Problem 6.3
Let U have a U(0,1) distribution. Show that
Z=1-U h
Z=1 U has a U(0,1) distribution by deriving th
U(0 1) di trib ti b d ri i the
probability density function or the distribut

Chapter 4
Ch
4:
Discrete Random Variables
Homework Solutions 4.2, 4.9, 4.11, 4.14
2
Problem 4.2
Let X be a discrete random variable with
probability mass function p given by:
a
-1
1
0
1
2
p(a)
1/4
1/8
1/8
1/2
and p(a) = 0 for all other a.
p( )
a. Let the

5
Continuous Random Variables
5.1 Probability Density Functions
_
Continuous random variables as a refinement of discrete random variables.
Imagine that the outcome '#)$ happens with probability :. Refining
towards a four decimal place accuracy, this prob

8
Computation with Random Variables
8.1 Transforming discrete random variables
_
There are many ways to make new random variables from old ones. Of
course this is not a goal in itself; usually new variables are created
naturally in the process of solving

9
Joint Distributions and Independence
9.1 Joint distributions of discrete random variables
_
Random variables often provide information about one another. In order
to capture this, we introduce the joint distribution of two or more random
variables.
For

10 Covariance and Correlation
10.1 Expectation and Joint distributions
_
In this chapter we see how the joint distribution of two or more random
variables is used to compute the expectation of a combination of these
random variables.
We discuss the expect

13 The Law of Large Numbers
13.1 Averages vary less . . .
_
For many experiments and observations concerning natural phenomena
such as measuring the speed of light performing the procedure twice
under (what seem) identical conditions results in two diffe

20 Efficiency and Mean Squared Error
20.2 Variance of an Estimator . . .
_
Call Center Example: Consider the arrivals of calls in a 24 hour call-center. One
is interested in the intensity average number of arrivals per hour) at which
calls arrive on a gen

19 Unbiased Estimators
19.1 Estimators . . .
_
Call Center Example: Consider the arrivals of calls in a 24 hour call-center. One
is interested in the intensity average number of arrivals per hour) at which
calls arrive on a generic day and the probability

14 The Central Limit Theorem
14.1 Standardizing Averages . . .
_
Expectation and Variance of an Average. If \ 8 is the average of 8
independent rv's with the same expectation . and variance 5# then
5#
I\ 8 . and Z +<\ 8
8
Example: Let \3 \ \ having a K+7

16 Exploratory Data Analysis: Numerical Summaries
16.0 Introduction . . .
_
The classical way to describe important features of a dataset is to give several
numerical summaries for:
1) The center of a dataset
2) The amount of variability among the element

23 Confidence Intervals for the Mean
23.1 General Principle . . .
_
We discussed sample statistics X \" \# \8 as estimators for
distribution features, where \3 are 33. and \3 \
If we have at our disposal an estimator X \" \8 for an unknown
parameter ), we

15 Exploratory Data Analysis: Graphical Summaries
15.0 Introduction . . .
_
We have focused on probability models to describe random phenomena.
Confronted with a new (uncertain) phenomenon, one conducts
experiments and records observations concerning the

7
Expectation and Variance
7.1 Expected Values
_
Random variables are complicated objects, containing a lot of
information on the experiments that are modeled by them.
Typically, random variables are summarized by two numbers:
The expected value: also cal

6
Simulation
6.1 What is simulation?
_
In simulation, one uses a model to create specific situations in order to
study the response of the model to them and then interprets this in terms
of what would happen to the system in the real world.
Purpose: Carry

4
Discrete Random Variables
4.1 Random Variables
_
Example: Snakes and Ladders, where the moves are determined by the sum
of two independent throws with a die.
H =" =# A" A# " # $ % & '
" " " # " ' # " & ' ' '
However, as we play the game we are only int

Chapter 3:
p
Conditional probability and
independence
Homework Solutions 3.6, 3.11 and 3.17
H m
rk S l ti n 3 6 3 11 nd 3 17
2
Problem 3.6
We choose a month of the year, in such a manner
year
that each month has the same probability. Find out
whether the

Chapter 2
Ch
2:
Outcomes, Events and Probability
Homework Solutions 2.3, 2.13 and 2.15
2
Problem 2.3
Let C and D be two events for which one
knows that P(C) = 0.3, P(D) = 0.4, and
P(C D) = 0 2 What is P(Cc D)?
0.2.
P(D) = P(Cc D) + P(C D)
0.4 = P(Cc D) +

Chapter 5
Ch
5:
Continuous random variables
Homework Solutions 5.1, 5.3, 5.10, 5.11
2
Problem 5.1
Let X be a continuous random variable with
probability density function
3 / 4
f ( x) = 1 / 4
0
for
f 0 x 1
for 2 x 3
elsewhere
l
h
a. Draw a graph of f.
g p

Chapter 7
Ch
7:
Expectation and variance
Homework Solutions 7.4, 7.7 and 7.13
2
Problem 7.4
Let X be a random variable with E[X] = 2
2,
Var(X) = 4. Compute the expectation and
variance of 3 2X
2X.
3
Problem 7.4
Given:
E[ X ] = 2, Var ( X ) = 4
Mean Value

Chapter 9
Ch
9:
Joint distributions and independence
Homework Solutions 9.1, 9.7 and 9.12
2
Problem 9.1
The joint probabilities P(X = a, Y = b) of discrete
random variables X and Y are given in the following
table. Determine the marginal probability dist

Chapter 8
Ch
8:
Computations with random variables
Homework Solutions 8.4, 8.13 and 8.16
2
Problem 8.4
Transforming exponential distributions
distributions.
a. Let X have an Exp() distribution. Determine
the distribution function of X. What kind of
X
dis

Chapter 11:
p
More computations with more random
variables
Homework Solutions 11 2 11.3 and 11.6
H m
rk S l ti n 11.2, 11 3 nd 11 6
2
Problem 11.2 (a)
Consider a discrete random variable X taking values k =
k
0, 1, 2, with probabilities
P(X = k) = e ,
k

Chapter 13:
The law of large numbers
Homework Solutions 13.1, 13.5 and 13.11
2
Problem 13.1
Verify the a few as you did in Quick
exercise 13.2 for the following distributions:
U(-1,1), U(-a,a),
N(0,1), N(,),
Par(3), Geo(1/2).
Construct a table as in the

Chapter 10:
Covariance and correlation
Homework Solutions 10.4, 10.8 and 10.15
2
Problem 10.4
Consider the joint probability distribution of the
discrete random variables X and Y from the
Melancholia Exercise 9.1. Compute Cov(X,Y).
a
b
1
2
3
4
1
16/136
3

Lecture Notes ApSc 3115/6115:
Engineering Analysis III
Chapter 2: Outcomes, Events, and Probability
Version: &"*#!"%
Text Book: A Modern Introduction to Probability and Statistics,
Understanding Why and How
By: F.M. Dekking. C. Kraaikamp, H.P.Lopuha and L

Lecture Notes ApSc 3115/6115:
Engineering Analysis III
Chapter 3: Conditional Probability and Independence
Version: 5/21/2014
Text Book: A Modern Introduction to Probability and Statistics,
Understanding Why and How
By: F.M. Dekking. C. Kraaikamp, H.P.Lop

Lecture Notes ApSc 3115/6115:
Engineering Analysis III
Chapter 1: Why Probability and Statistics?
Version: !&"*#!"%
Text Book: A Modern Introduction to Probability and Statistics,
Understanding Why and How
By: F.M. Dekking. C. Kraaikamp, H.P.Lopuha and L.