APPM 2360, Summer 2016: Exam 2
June 24, 2016
Instructions: Please show all of your work and make your methods and reasoning clear. Answers out
of the blue with no supporting work will receive no credit (unless the directions to a specific problem say
othe
APPM 2360, Summer 2016: Exam 3
July 8, 2016
Instructions: Please show all of your work and make your methods and reasoning clear. Answers out
of the blue with no supporting work will receive no credit (unless the directions to a specific problem say
other
APPM 2360, Summer 2016: Exam 4
July 22, 2016
Instructions: Please show all of your work and make your methods and reasoning clear. Answers out
of the blue with no supporting work will receive no credit (unless the directions to a specific problem say
othe
APPM 2360: Midterm exam 2, SOLUTIONS
October 19, 2016
Problem 1: (24 points) True/False (answer True if it is always true; otherwise answer False)
No justification is needed.
a b c
1
(a) The system of equations
~x =
has at least one solution ~x for any
APPM 2360: Midterm Exam 3 [SOLUTIONS]
November 16, 2016
Problem 1: (30 points; 6 points each) True/False questions. Answer True if the statement is
always true. Answer False otherwise. Box your answer. No partial credit will be given.
(a) If y(t) 6= 0 is
APPM 2360: Final Exam
December 13, 2016
Problem 1: (30 points) True/False questions. Answer True if the statement is always true,
otherwise answer False. Box your answer. No partial credit will be given.
(a) The solution space of 3y 000 + y 00 y = 0 is a
Solution: APPM 2360
Exam1
Fall 2016
Problem 1: (36 points, 6 points each) True/False (answer True if it is always true otherwise
answer False) or Short Answer for the following problems. No justification is needed.
(a) The differential equation y 0 + y si
APPM 2360, Summer 2016: Exam 1
June 10, 2016
Instructions: Please show all of your work and make your methods and reasoning clear. Answers out
of the blue with no supporting work will receive no credit (unless the directions to a specific problem say
othe
It assumes that the reader has a good knowledge of several Calculus II topics including some
integration techniques, parametric equations, vectors, and knowledge of three dimensional space.
Here are a couple of warnings to my students who may be here to g
1. With that being said I will, on occasion, work problems off the top of my head when I can to
provide more examples than just those in my notes. Also, I often dont have time in class to
work all of the problems in the notes and so you will find that som
1. notes fairly close as far as worked problems go. With that being said I will, on occasion, work
problems off the top of my head when I can to provide more examples than just those in my
notes. Also, I often dont have time in class to work all of the pr
1. In general I try to work problems in class that are different from my notes. However, with
Calculus III many of the problems are difficult to make up on the spur of the moment and so in
this class my class work will follow these notes fairly close as f
1. of the problems in the notes and so you will find that some sections contain problems that
werent worked in class due to time restrictions.
2. Sometimes questions in class will lead down paths that are not covered here. I try to anticipate
as many of t
1. Also, I often dont have time in class to work all of the problems in the notes and so you will find
that some sections contain problems that werent worked in class due to time restrictions.
2. Sometimes questions in class will lead down paths that are
An-Najah National University
Engineering College
Civil Engineering Department
Strength of Material
Laboratory Manual
Prepared by:
Dr. Isam Jardaneh
Eng. Hamees Tubeileh
2009
An- Najah National University
Civil Engineering Department
Mechanics of Material
1 Introduction to Operations Research
In its first section, this introductory chapter first introduces operations research as a discipline. It defines its function and then traces its roots to its beginnings. The second section highlights some of the main
APPM 4/5520 Problem Set Nine (Due Wednesday, November 30th) 1. Let X1 , X2 , . . . , Xn be a random sample from the exponential distribution with rate . (a) Find the MLE (maximum likelihood estimator) for the "tail probability" P (X1 > x) for some fixed x
APPM 4/5520 Problem Set Eight (Due Wednesday, November 16th) 1. Let X1 , X2 , . . . Xn be a random sample from the binomial distribution with parameters m and p. Find the MLE of P (X = 0). 2. Let X1 , X2 , . . . , Xn be a random sample from the P oisson()
APPM 4/5520 Solutions to Problem Set Seven 1. The first population moment is
1 = E[X] =
x e-(x-) dx = + 1.
(Note that since the pdf is that of an exponential with rate 1 that has been shifted by to the right, the mean is 1 (the mean of the exponential wit
APPM 4/5520 Problem Set Seven (Due Wednesday, November 9th) 1. Consider the "shifted" rate 1 exponential distribution with pdf f (x) = e-(x-) I(,) (x). This distribution is frequently encountered in reliability and product lifetime analyses. The new param
APPM 4/5520 Solutions to Problem Set Six 1. Recall that we define a t-distribution with n degress of freedom by letting Z N (0, 1) and W 2 (n) be independent random variables. Let T = Then T t(n). For this problem, note that X1 X1 = = |X2 | 2 X2 X1 / = (X
APPM 4/5520 Problem Set Six (Due Wednesday, October 19) 1. Let X1 , X2 be a random sample from the N (0, 2 ) distribution. Find the distribution of X1 ? |X2 | Name it! (Hint: |X2 | =
2 X2 .)
2. Let X(1) , X(2) , . . . , X(n) denote the order statistics of
APPM 4/5520 Solutions to Problem Set Five 1. (a)
n i=1 (xi
- )2 = = =
n i=1 (xi n i=1
- x + x - )2
(xi - x)2 + 2(xi - x)(x - ) + (x - )2 - x)2 + 2(x - )
n n n i=1 (xi
n i=1 (xi
- x) +
n i=1 (x
- )2
For the middle term, note that
n i=1 n
(xi - x) =
n i=1
i
APPM 4/5520 Problem Set Five (Due Wednesday, October 12th) 1. Suppose that X1 , X2 , . . . , Xn N (, 2 ). (a) Show that
n 2 i=1 (xi - ) 2 n i=1 (xi - 2 iid
=
x)2
+
n(x - )2 . 2
(b) Write down the joint pdf for X1 , X2 , . . . , Xn and use part (a) to rewr
APPM 4/5520 Problem Set Four (Due Wednesday, October 5th) 1. Let X1 , X2 , . . . , Xn be a random sample from any distribution with mean and variance 2 . From class, we know that = X is an unbiased estimator of . ^ Suppose that we want to estimate the var
APPM 4/5520 Solutions to Problem Set Three 1. (a) f (x) = p(1 - p)x Icfw_0,1,2,. (x) MX (t) = E[etX ] = = p
x=0 tx x=0 e
p(1 - p)x
x
et (1 - p)
This geometric sum converges if et (1 - p) < 1. (Note that it is already positive since et > 0 and since 0 p
APPM 4/5520 Problem Set Three (Due Wednesday, September 14th) 1. Let X geom0 (p). (a) Compute the moment generating function of X. Be sure to give and explain restrictions, if any, on the domain of the mgf. (b) Let Y geom1 (p). How does Y relate to X? Use
APPM 4/5520 Problem Set Two (Due Wednesday, September 7th) 1. Suppose that X1 , X2 exp(rate = ). Find the distribution of 2. Let X1 and X2 have the joint pdf fX1 ,X2 (x1 , x2 ) = 2e-x1 -x2 I(0,x2 ) (x1 )I(0,) (x2 ). (a) Find the marginal pdfs for X1 and X