Chapter 4: Probability The Study of Randomness
4.1: Randomness
4.2: Probability Models
4.3: Random Variables
4.4: Means and Variances of Random Variables
4.5: General Probability Rules
4.1: Randomness
A phenomenon is random if individual outcomes are
unce
4.4: Means and Variances of Random Variables
Example 2: Compare the means of the random variables X and Y :
X
Probability
2
0.4
3
0.6
Y = 2X
Probability
4
0.4
6
0.6
4.4: Means and Variances of Random Variables
Example 3: Linda is a sales associate at an a
4.3: Random Variables
How do we assign probabilities to events involving a continuous
random variable X ?
Example: Let X be the amount of time (in minutes) that you will
have to wait for a friend who will be at most 15 minutes late.
Then, X is a continuou
Chapter 4: Probability The Study of Randomness
4.1: Randomness
4.2: Probability Models
4.3: Random Variables
4.4: Means and Variances of Random Variables
4.5: General Probability Rules
4.5: General Probability Rules
Objectives
Venn diagrams
General additi
4.5: General Probability Rules
Conditional Probability
Notation: P(AB) = the probability of A given that event B has
occurred.
P(AB) =
number of outcomes in A and B
number of outcomes in B
Example 1: Consider the current U.S. Senate membership:
Female
M
4.5: General Probability Rules
Recall: Events A and B are independent if they have no influence
on each others occurrence.
If A and B are independent, then P(A and B) = P(A)P(B).
If A and B are independent, then P(BA) = P(B).
Example: Supposed two dice a
4.4: Means and Variances of Random Variables
Mean of a Continuous Random Variable
Recall: The probability distribution of continuous random variables
is described by a density curve.
The mean lies at the center of symmetric density curves such
as the norm
MAT 221  Fall 2016
Formulae
If X is a discrete random variable taking on values x1 , . . . , xk with respective probabilities
p1 , . . . , pk , then the mean is X = x1 p1 + + xk pk , the variance is
q
2
2 .
X
= (x1 X )2 p1 + + (xk X )2 pk , and the stan
4.2: Probability Models
Exercise 4.29:
Exercise 4.30:
Ex 3: Suppose two people are randomly selected from the world
population. Let E be the event that both of these people live in
Europe. Which of the following events is Ec ?
(a) The event that neither o
MAT 532 HOMEWORK 1.5
DUE TUESDAY
13 SEPTEMBER 2016 AT THE BEGINNING OF CLASS
1. (a) Using 3digit floating point arithmetic, apply Gaussian elimination (without partial
pivoting) to solve the following system.
"
103 1
2
3
2
44
#
.
Dont solve with backsub
MAT 532 HOMEWORK 3.10
DUE TUESSDAY
25 OCTOBER 2016 AT THE BEGINNING OF CLASS
1. Let
1
4
5
A=
4 18 26 .
3 16 30
(a) Find the LU factorization of A .
(b) Use your LU factorization to solve Ax = b, where b =
(c) Use your LU factorization to determine A
1
h
6
MAT 532 HOMEWORK 1.6
DUE THURSDAY
15 SEPTEMBER 2016 AT THE BEGINNING OF CLASS
1. Use geometry to decide which of the following three linear systems is worstconditioned
and which is bestconditioned. Explain your answer.
(
1.001 x
y = .235
x + .0001 y =
MAT 532 HOMEWORK 1.4
DUE THURSDAY
8 SEPTEMBER 2016 AT THE BEGINNING OF CLASS
1. (a) Set up the linear system to approximate a solution to the boundary value problem
y00 ( t) = kt
on the interval [0, 1], with boundary conditions y(0) = 1 and y(1) = 0, usin
MAT 532 HOMEWORK 1.2, 1.3
DUE TUESDAY
6 SEPTEMBER 2016 AT THE BEGINNING OF CLASS
1. Use Gaussian elimination with backsubstitution to solve the system. You should apply the algorithm as discussed in class; full credit will be given only for complete full
MAT 532 HOMEWORK 3.10
DUE TUESDAY
25 OCTOBER 2016 AT THE BEGINNING OF CLASS
1. Let
1
1
3
12
1 2 8
.
1 1
4 16
A=
1
(a) Find a nonsingular matrix P such that P A is the RREF of A .
(b) Find nonsingular matrices P and Q such that P AQ is the rank normal form
MAT 532 HOMEWORK 4.2
DUE THURSDAY
1. Given
27 OCTOBER 2016 AT THE BEGINNING OF CLASS
1
1
3
12
1 2 8
,
1 1
4 16
A=
1
find nonredundant spanning sets for:
(a) R ( A )
(b) R ( A T )
(c) N ( A )
(This is the same A as in the HW for 3.9, so you can use your R
Vector Calculus
3.1 Introduction
3.2 Velocity and acceleration
3.2 Derivatives of a scalar point function
3.3 The gradient of a scalar point function
3.4 Exercises
3.5 Derivatives of a vector point function
3.6 Divergence of a vector field
3.7 Curl of a v
Except where otherwise noted in the source code (e.g. the files hash.c,
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* Copyright (C) 19982001 Bjorn Reese and Daniel Veillard.
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MAT 295 Quiz 2.62.7 Name: “5.)
1. [6 pts] ( Based on problem from Spring 2010 Final Exam) A hot air balloon rising straight up from a
level ﬁeld is tracked by a camera on the ground 500 feet from the lift off point. When the camera’s
angle of elevation i