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Applied Math 5600
Bengt Fornberg
Final
December 15, 2016
7.30  10.00 pm
Return this problem set as the first part of your solution set.
Name: _

1.
The figure to the right illustrates (and
gives some facts about) the cubic Bspline for a unitspaced gri
Homework 8 Solutions APPM 4440 Fall 2016
1. (4.4.3) (a) Using f (t) = t2 and g(t) = t3 , then
f (1) f (0)
f 0 (c)
2c
2
=1= 0
= 2 = c1
g(1) g(0)
g (c)
3c
3
Thus c = 23 .
(b) On the other hand for f (1) f (0) = 1 = f 0 (c)(1 0) = 2c, we have c = 21 uniquely
Homework 9 Solutions APPM 4440 Fall 2016
1. Problem 1: Go over the statements of your definitions and theorems one more time. Then assign
yourself 10 points for this problem.
2. (6.1.1c) Since f (x) = x2 is monotone decreasing on [0, 1], the infima occur
Homework 7 Solutions APPM 4440 Fall 2016
1. (4.1.7) We assume f : R R is differentiable at x0 = 1, and the result of problem 4.1.6
(a) Let g(x) = x + 1, and x0 = 0 then
A = lim
h0
(b) Let g(t) =
f (1 + h) f (1)
f (g(h) f (g(0)
= lim
= f 0 (g(0) = f 0 (1).
Homework 1 Solutions APPM 4440 Fall 2016
Directions: Grade your homework and turn in the graded homework in class on Friday, Sept 2.
Each problem is worth 5 points, so this hw is worth 50 points total. Use a different color pen for
your grading and be sur
Homework 5 Solutions APPM 4440 Fall 2016
1. (3.1.1) These problems are very close to 2.1.1.
(a) FALSE. Consider the functions f, g : R R defined by f (x) = 1, x 0, and f (x) = 1, x <
0, and g(x) = f (x). Then f + g = 0, but neither f nor g is continuous.
Homework 6 Solutions APPM 4440 Fall 2016
1. (3.4.7) We are given f : D R and g : D R are uniformly continuous and bounded. We need
to show that the product, f g : d R is also uniformly continuous.
Proof Assume that cfw_un and cfw_vn are sequences in D w
Homework 10 Solutions APPM 4440 Fall 2016
1. Suppose a function E : R R is defined as the solution of the ODE
E 0 (x) = xE(x),
E(0) = 1.
We will assume that this equation has a solution, and that E(x) 6= 0 for all x R. For this
problem, you are to answer
Homework 2 Solutions APPM 4440 Fall 2016
Directions: Grade your homework and turn in the graded homework in class on Friday, Sept 9. (If
you need extra time, thats fine. You can put your graded homework under my office door later
on Friday or sometime on
Homework 3 Solutions APPM 4440 Fall 2016
1. (2.1.1) (Grading: 1 point each for parts a, b, and d. 2 points for part c.)
(a) FALSE. Consider the sequence an = (1)n . Since cfw_a2n = cfw_1 this converges to 1,
but cfw_an does not converge, recall Example
Homework 4 Solutions APPM 4440 Fall 2014
1. Copy the definitions from the text. There are many possible examples for each definition. (Grading: This problem is worth 5 points. If you have questions about your examples, please ask.)
2. (2.3.1)
(a) FALSE. O
APPM/MATH 4/5520
Exam II Review Problems
The exam will be on Thursday, November 10th from 6:30 to 9:00pm in FLMG 155.
Optional Extra Review Session: will be on Wednesday, November 9th from 6 to 8 pm in MUEN
E113.
1. Let X1 , X2 be a random sample from the
APPM 4/5520
Problem Set Ten (Due Wednesday, November 9th)
1. Let X1 , X2 , . . . , Xn be a random sample from the exp(rate = ) distribution. Verify that
P
S = ni=1 Xi is a sufficient statistic from the definition of sufficiency. (ie: do not use the
Factor
APPM/MATH 4/5520
Solutions to Problem Set One
c
1. (a) F (x) = 1 e(x)
on [0, ) and is 0 otherwise So,
c
f (x) = F 0 (x) = 0 [c(x)c1 e(x) ]
c)
= cc xc1 e(x)
and f is zero otherwise.
All together,
c
f (x) = cc xc1 e(x) I(0,) (x).
This is the pdf for the
Wei
APPM/MATH 4/5520
Exam I Review Problems
Exam I: Thursday, September 9th from 6:30 to 9pm in a room TBA.
Optional Extra Review Session: Wednesday, September 28th from 6 to 8 pm in a room TBA.
(Note: I reserved these rooms a long time ago but just discovere
APPM/MATH 4/5520
Problem Set Two (Due Wednesday, September 7th)
1. Suppose that U is a continuous random variable that is uniformly distributed on the interval
(1, 1). That is, U unif (1, 1).
Let > 0 and let
Y =
2
1U
1.
Find the distribution of Y . (Name
APPM 4/5520
Problem Set Seven (Due Wednesday, October 19th)
1. (a) On HW 5, Problem 1, we saw that for a random sample X1 , X2 , . . . , Xn from a distribution with continous and invertible cdf F ,
2
n
X
ln F (Xi ) 2 (2n).
i=1
Recall that the proof was ba
APPM/MATH 4/5520
Problem Set Four (Due Wednesday, September 21st)
1. Let X1 , X2 , . . . , Xn be a random sample from the P areto() distribution. (By default, the
Pareto distribution refers to the continuous Pareto distribution.)
(a) Find the distribution
APPM 4/5520
Problem Set Eight (Due Wednesday, October 26th)
1. Let X1 , X2 , . . . , Xn be a random sample from the distribution with pdf
f (x; ) =
(2) 1
x (1 x)1 I(0,1) (x).
[()]2
Find the method of moments estimator (MME) of .
2. Let X1 , X2 , . . . , X
APPM 4/5520
Problem Set Five (Due Wednesday, September 28th)
1. Suppose that X1 , X2 , . . . , Xn is a random sample from a distribution with pdf f and cdf F .
Show that
n
2
X
ln F (Xi ) 2 (2n).
i=1
2. Let X1 , X2 , . . . , Xn be a random sample from any
APPM 4/5520
Solutions to Problem Set Eleven
iid
1. First note that X1 , X2 , . . . , Xn P oisson() implies that S =
is most easily shown using moment generating functions.)
P
Xi P oisson(n). (This
In order to show that S is sufficient by the definition, w
APPM/MATH 4/5520
Solutions to Exam I Review Problems
1. (a)
fX1 (x1 ) =
=
R
fX1 ,X2 (x1 , x2 ) dx2
R
x1
2ex1 x2 dx2
= 2e2x1
x1 was below x2 , but when marginalizing out x2 , we ran it over all values from 0 to
and so there was no upper bound on x1 . The
APPM/MATH 4/5520
Exam II Review Problems
The exam will be on Thursday, November 12th from 6:30 to 9:00pm in FLMG 155.
Optional Extra Review Session: will be on Wednesday, November 13th from 6 to 8 pm in a
room TBA.
1. Let X1 , X2 be a random sample from t
APPM/MATH 4/5520
Exam II Review Problem Solutions (123)
Important Note: Do not count on being able to make corrections for this exam. That was a one
time opportunity for Exam I to help bridge the gap that some people had with prerequisites!
1. Recall tha
APPM/MATH 4/5520
Exam I Review Problems
Exam I: Thursday, October 1st from 6:30 to 9pm in MATH 100.
Optional Extra Review Session: Wednesday, September 30th from 6 to 8 pm in ECCR 108.
The actual exam will have 6 problems and you will only have to do 5 of