Chapter 1
Introduction
The introduction of the course is broken into two sections. The rst section describes
the importance of friction and wear to today's engineer and introduces some of the
basic concepts that will be pursued later in more detail. The s

A-AE 567
Quiz
Spring 2011
Clearly write your nal answer on the exam.
NAME:
1
Problem 1. Consider the random variable x given by the distribution:
Fx (x) = ex
if x < 0
= ex
Here and are constants. Then compute
x = E x =
2
x = E (x x )2 =
2
if x 0.
Problem

A-AE 567 Quiz Spring 2010 Solutions
Problem 1. Assume that x is uniform over cfw_x : 1 |x| 2, that is, fx (x) = =0 Let y = x2 . Compute: 7 E y = E x2 = . 3 Notice that = 1/2. Thus 1 22 1 1 2 2 2 x dx + x dx Ey = Ex = x fx (x)dx = 2 2 21 x3 = 6
1
if 1 |x|

A-AE 567
Quiz
Spring 2010
Clearly write your nal answer on the exam.
NAME:
1
Problem 1. Assume that x is uniform over cfw_x : 1 |x| 2, that is, fx (x) = =0 Let y = x2 . Compute Ey = if 1 |x| 2 otherwise.
Find the density fy (y ) =
2
Problem 2. Assume that

A-AE 567
Quiz
Spring 2009
Clearly write your nal answer on the exam.
NAME:
1
Problem 1. Assume that x and y are independent uniform random
variables over the interval [0, 1]. Let z be the random variable dened
by z = |y x|. Hint you may need Fz (z ). Then

A-AE 567
Quiz
Spring 2009
Clearly write your nal answer on the exam.
NAME:
1
Problem 1. Assume that x and y are independent uniform random
variables over the interval [0, 1]. Let z be the random variable dened
by z = |y x|. Hint you may need Fz (z ). Then

Notes on Probability
A.E. Frazho
2
Contents
1 The
1.1
1.2
1.3
1.4
1.5
Probability Measure
The axioms of probability . . . . .
Conditional probability . . . . . . .
The birthday problem . . . . . . .
The gamblers ruin problem . . . .
Bayes Rule . . . . . .

A-AE 567 Final Homework Spring 2011
You will need Matlab and Simulink.
You work must be neat and easy to read.
Clearly, identify your answers in a box.
You will loose points for poorly written work.
You must work alone.
Exam is due in a box outside my oce

A-AE 567 Final Homework Spring 2010
You will need Matlab and Simulink.
You work must be neat and easy to read.
Clearly, identify your answers in a box.
You will loose points for poorly written work.
You must work alone.
Due 4:00 pm Thursday May 6 at the b

A-AE 567
Quiz
Spring 2011
Clearly write your nal answer on the exam.
NAME:
1
Problem 1. Consider the random variable x given by the distribution:
Fx (x) = ex
if x < 0
= ex
if x 0.
Here and are constants. Then compute
x = E x = 0
2
x = E (x x )2 = E x2 =