Math 544 Exam 4 Summer 2000
Use the paper provided. Put your name on the front of the rst page and the back
of the last page. Problem 7 is worth 8 points each. The other problems are worth
7 points ea
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Quiz for May 30, 2006
Let A and B be nonempty subsets of positive real numbers that are bounded from
above. Let C = cfw_ab  a A and b B . Prove that sup C = (sup A)(sup B ) .
ANSWER
Math 554, Exam 1, Summer 2006
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Leave room on the upper left hand corner
of each page for
Notes on The Final Exam, Math 554, Summer 2006
1. The nal exam is Thursday, June 29 in our usual room at our usual time. The
exam is comprehensive, worth 100 points. All theorems and denitions that yo
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Quiz for June 14, 2006
a. Let cfw_O  A be a set of open subsets of R . Prove that
O is an
A
open subset of R .
b. Give an example of a set of open subsets cfw_O  A of R with
O not
A
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Quiz for June 15, 2004
Let cfw_an and cfw_bn be Cauchy sequences. Prove that the sequence cfw_an bn is also
a Cauchy sequence.
ANSWER: Let > 0 be arbitrary, but xed.
The sequence
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Quiz for June 6, 2005
Let cfw_pn be a bounded sequence of real numbers and let p R be such that
every convergent subsequence of cfw_pn converges to p . Prove that the sequence
cfw_p
Math 554, Exam 2, Summer 2006 Solutions
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Leave room on the upper left hand corner
of eac
Math 554, Exam 2, Summer 2006
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Leave room on the upper left hand corner
of each page for
Math 554, Exam 1, Summer 2006
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Leave room on the upper left hand corner
of each page for
Math 554, Final Exam, Summer 2006
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Leave room on the upper left hand corner
of each page
Math 554, Final Exam, Summer 2006
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Leave room on the upper left hand corner
of each page
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Quiz for June 8, 2004
Prove that if the sequence cfw_an converges to a , then the sequence cfw_an 
converges to a . Is the converse true? Prove or give a counter example.
ANSWER:
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Quiz for June 5, 2006
2
+1
Give a complete proof that the limit of the sequence cfw_ n2n2 is 1 . In other words,
2
given > 0 , you are required to nd a formula for n0 with the proper
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Quiz for June 1, 2006
Let X , Y , and Z be sets. Suppose that f : X Y and g : Y Z are
onetoone functions. Prove that the function g f : X Z is onetoone.
ANSWER: Let x1 and x2 be e
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Quiz for June 22, 2006
Let f and g be functions from the subset E of R to R , and let p be a limit
point of E . Suppose that lim f (x) exists and equals A . Suppose, also, that
xp
lim
I The Most Commonly Used Graphical Interpretation of Failure Rate
Let a random variable X be a mean time to failure of any system. Then, a typical form of
failure rate is shown in the following gure.
3.5. Failure Rate
Let X be a nonnegative random variable with continuous distribution. Then, the failure rate
for F(t) < l is defined as:
The following formula can easily be derived:
MI): lim P(t 4
4.2. Marginal Distribution
Denition
Let X = (XI, X2. . X.1 ) be a random vector. The random vector Y = (1111.111.2 . XI: ). vvhere
k < n, if. e cfw_ 1,2,.,n , in i i, pro at i v, is called the margina
A group of all values cfw_2: = X(co). a) e Q is called sample space.
3.2. Distribution Function
Denition: The distribution function of a random variable X is F ( I) and for each IE R it has
the value:
4. RANDOM VECTOR
Study time: 60 minutes
@
Learning Objectives  you will be able to
t Describe a random vector and its joint distribution
 Explain the concepts of marginal and conditional probability
Example
Logistic probability distribution has the following distribution function F (x) and probability
density x):
'cfw_n+131x
Fun; x = 3?
1+ erswsx) ( 1 + g(Jrlx) )2
0.75
0.5 F(x)
0.25
0.4
0.3
0.2
For a distribution function of a discrete random variable the following is true:
F(x)=ZP(X=xE)
Example
Throwing a dice. X a number of dots obtained
U6
3.4. Continuous Random Variable
If a random v
If you know the distribution function you can easily determine the probability density
function and vice versa.
The area below the for) curve for x as o;b ); cfw_ 1:1,!) E R ) in any interval is the p
discrete RV: y. = 2 [xi  EXF H Xi)
i
cfw_1
continuous RV: ,u. = I (x  EX )r.f(x ) ch:
C
if stated progression or integral converge absolutely.
2. Expected Value (Mean) EX = pf
discrete RV: EX 2 2 x
a3 :2 0 positively skewed set
6. Kurtosis a4 = ,ua Er:
Is a level of kurtosis (atness):
a4 = 3 normal kurtosis (i.e. kurtosis of normal distribution)
21.; < 3 lower kurtosis than kurtosis of normal di