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Spring 2015
Math 461 B/C, Spring 2009
Midterm Exam 3 Solutions and Comments
1. Let X have normal distribution with mean 1 and variance 4.
10 pts
(a) Find P (X 1).
Solution.
P (X 1) = P (1 X 1)
X 1
1 1
11
=P
2
2
2
= (0) (1) = (0) (1 (1)
= 0.5 (1 0.84) = 0.34
10 pt
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Spring 2015
Prof. Marco Panesi
AE433, Fall 2016
HW #1
Due September 7, 2016 by 5 pm.
Problem 1
Answer the following questions:
(a) What is an airbreathing propulsion system, and what is the difference between airbreathing and
nonairbreathing systems?
(b) List few
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Spring 2015
Math 461 Section P1
Quiz 5 Solution
Feb 28, 2013
2. A salesman has scheduled two appointments to sell encyclopedias. His
first appointment will lead to a sale with probability 0.3, and his second
appointment will lead independently to a sale with probabil
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Spring 2015
Math 461, Solution to Written Homework 8
1. The joint density function of X and Y is given by
(
24xy, x (0, 1), y (0, 1), 0 < x + y < 1,
f (x, y) =
0,
otherwise .
Are X and Y independent? Find E[X].
Solution For x (0, 1),
Z
fX (x) =
1x
24xydy = 12x(1 x)2
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Spring 2015
Math 461
Midterm #1  Solution to the practice exam
1. (a)
i. True (definition)
ii. True: The equation A~x = ~b is equivalent to the vector equation x1 a~1 + +xn a~n =
~b.
iii. False: The equation A~x = ~b is consistent if and only if the augmented matrix
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Spring 2015
Math 461
Final  Practice exam
This is a practice exam. The actual exam may include material that is not on this practice
exam and the wording of questions may be different. Make sure that you do all suggested homework
problems and review all the material
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Spring 2015
Math 461 F
Homework 3 Solutions
Spring 2011
Drew Armstrong
Problems.
p
A.1. How many different (complex) numbers does the expression 1 + 3 represent?
Find a polynomial over Z which has these numbers as its roots.
The symbol 3 represents two numbers and th
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Spring 2015
Math 461  Linear Algebra for Scientists and Engineers
MATLAB Assignment #4
Eigenvalues and Eigenvectors If A is a square nn matrix, then the command eig(A)
will produce a vector whose entries are the n eigenvalues of A (including multiplicities). If
2 is
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Spring 2015
Math 461 C, Spring 2009
Final Exam Solutions and Comments
1. The following parts are independent of each other.
10 pts
(a) What is the coefficient of x9 y 10 z 11 if (x + y + z)30 is expanded? Express this coefficient in terms of
powers and/or factorials
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Spring 2015
Math 461 B/C, Spring 2009
Midterm Exam 2 Solutions and Comments
1. Assume A and B are independent events with P (A) = 0.2 and P (B) = 0.3. Let C denote the event
that none of the events A and B occurs, and let D be the event that exactly one of the events
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Spring 2015
Homework 1: Solutions
1. Consider how the de Broglies suggestion might explain some properties of the hydrogen atom.
a. Show that the assumption
h
p = m =
and the quantization condiction that the length of a circular orbit be an integer multiple of the le
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Spring 2015
Prof. Daniel J. Bodony
AE433, Fall 2015
AE433Aerospace Propulsion : Fall 2015
Website http:/acoustics.ae.uiuc.edu, click on AE 433 on left.
Instructor Prof. Daniel J. Bodony (AE)
Office
Email
Office hours
313 Talbot
bodony@illinois.edu
TBD
Credit
Time
Lo
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Spring 2015
MATH 461/661 Homework 5
Solutions
1. 3.3.1(a) Let X1 and X2 be the numbers on the two balls drawn, then X = max(X1 , X2 ). The
sample space for the random variables X1 and X2 is S = cfw_1, 2, 3, 4, 5. The sample space for
RV X is S 0 = cfw_2, 3, 4, 5 sinc
Science and Aviation/Aerospace Technology in Society
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Spring 2015
Math 461 B/C, Spring 2009
Midterm Exam 1 Solutions and Comments
1. Suppose A, B and C are events with P (A) = P (B) = P (C) = 1/3, P (AB) = P (AC) = P (BC) = 1/4
and P (ABC) = 1/5. For each of the following events, first express the event in settheoretic
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Spring 2015
Math 461 Section P1
Quiz 4 Solution
Feb 13, 2013
1. An urn contains 6 white and 9 black balls. If 4 balls are to be randomly
selected without replacement, what is the probability that the first 2
selected are white and the last 2 black?
Solution. If Wi (B
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Spring 2015
Math 461 Section P1
Quiz 4 Solution
Feb 13, 2013
1. An urn contains 6 white and 9 black balls. If 4 balls are to be randomly
selected without replacement, what is the probability that the first 2
selected are white and the last 2 black?
Solution. If Wi (B
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Spring 2015
Math 461 Section P1
Quiz 6 Solution
March 7, 2013
1. You have $1000, and a certain commodity presently sells for $2 per ounce.
Suppose that after one week the commodity will sell either $1 or $4 per
ounce, with these two possibilities being equally likely
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Spring 2015
Math 461  Linear Algebra for Scientists and Engineers
MATLAB Assignment #3
The rank of a matrix: If A is a matrix, the command rank(A) computes the rank of
A. Because of roundoff errors, it is difficult to compute the rank (if an entry is 1016 , is it
a
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Spring 2015
Math 461 Section P1
Quiz 3 Solution
Feb 6, 2013
1. Sixty percent of students at a certain school wear neither a ring nor a
necklace. Twenty percent wear a ring and 30 percent wear a necklace.
If one of the students is chosen randomly, what is the probabil
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Spring 2015
Math 461 Section P1
Quiz 2 Solution
Jan 31, 2013
1. Consider a function f (x1 , x2 , . . . , xn ) of n variables at least r times differentiable. How many different partial derivatives of order r does f
process?
Solution: Assuming that the derivative with
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Spring 2015
Math 461 Section P1
Quiz 9 Solution
April 25, 2013
If X and Y are independent and identically distributed with mean and
variance 2 , find
E[(X Y )2 ].
Solution: We know that
E[Y 2 ] = E[X 2 ] = Var(X) + (E[X])2 = 2 + 2
and by independence
E[XY ] = E[X]E[Y
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Spring 2015
Math 461  Linear Algebra for Scientists and Engineers
MATLAB Assignment #2
More matrix operations:
If A is a matrix with real entries, then A is the transpose of A (if A has complex
entries then A is the conjugate transpose).
If A is a square invertibl
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Spring 2015
Math 461
Midterm #2  Solution to the practice exam
1. (a) True or False. Justify carefully your answer.
0
1
i. True: H = span 0 , 1 is a vector space.
0
0
ii. False: ~0 is not in S.
iii. False (the Rank Theorem gives dim(Nul(A)= 4)
iv. False: Only on
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Spring 2015
MATH 461/661 Homework 5
Solutions
1. 3.3.1(a) Let X1 and X2 be the numbers on the two balls drawn, then X = max(X1 , X2 ). The
sample space for the random variables X1 and X2 is S = cfw_1, 2, 3, 4, 5. The sample space for
RV X is S 0 = cfw_2, 3, 4, 5 sinc
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Spring 2015
Spring 2013  Math 461
Midterm #2  Solution
1. (a)
(b)
2. (a)
i.
ii.
iii.
iv.
True (such a matrix is invertible).
False (only if ~b = ~0).
True (definition of the kernel of the linear transformation).
True (if A is not invertible, then A~x = ~0 has a non
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Spring 2015
Math 461
Midterm #2  Practice exam
This is a practice exam. The actual exam may include material that is not on this practice exam and the wording of questions may be different. Make sure that you do all suggested
homework problems and review all the mat
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Spring 2015
Math 461
Practice exam #3  Solution
1. (a) False (the zero matrix is diagonalizable)
(b) False (the identity matrix is diagonalizable but its only evalue is 1)
(c) True (theorem)
(d) True (theorem  this is the normal equation)
2. (a) A~v1 = 9~v1 , so 1
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Spring 2015
Spring 2013  Math 461
Midterm #3  Solutions
1. (a) [10pts]
i.
ii.
iii.
iv.
True (theorem)
False (only if the column of the matrix are linearly independent)
True (theorem)
False (only similar matrices have the same eigenvalues  for instance any invertib
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Spring 2015
Math 461  Linear Algebra for Scientists and Engineers
MATLAB Assignment #1
This project is very simple. Its goal is to introduce you to Matlab. For an introduction
to Matlab, you can consult the introduction posted on the class website. Whenever you
use
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Spring 2015
Math 461 Section P1
Quiz 1 Solution
Jan 24, 2013
1. In how many ways can 3 novels, 2 mathematics books and a chemistry
book be arranged on a shelf if:
(a) the books can be arranged in any order?
(b) the mathematics books must be together and the novels mu
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Spring 2015
Math 461 Section P1
Quiz 8 Solution
April 11, 2013
1. The joint probability density function of X and Y is given by
6 x2 + xy
0 < x < 1, 0 < y < 2,
f (x, y) = 7
2
0
otherwise
(i) (10 points) Find P (X < Y ).
1
1
(ii) (15 points) Find P Y < X <
2
2
Sol
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Spring 2015
Spring 2013  Math 461
Midterm #3  May 6th
Write your name and section number on every page. (You only need to write and sign
the honor pledge on the first page).
This exam has 4 questions. Do each question on a different answer sheet.
Show enough wor