Vorbereitung zur Mathematik fr VWL und Statistik
Reinhard Ullrich Version 30.09.2010
Basierend auf: H. Schichl, R. Steinbauer,
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Inhaltsverzeichnis
1 Einleitung
Einfhrung in das mathematische
, Springer, 2009
2
2
Aussagenlogik
4
2.1 2.2
Aussagen -
Homework 2 Solutions
Math 171, Spring 2010
Please send corrections to [email protected]
9.6. Prove that if A and B are countable sets, then A B is countable.
Solution. For a fixed a A, let Ba = cfw_(a, b) A B | b B. Since B is countable, each
Ba is
Homework 3 Solutions
Math 171, Spring 2010
Please send corrections to [email protected]
17.4. Let cfw_an be a sequence with positive terms such that limn an = L > 0. Let x be a real number.
Prove that limn axn = Lx .
Solution. Let > 0. By Theorem
6
Completeness
6.1
Cauchy sequences
Definition 6.1. A sequence (xn ) of elements of a metric space (X, %) is
called a Cauchy sequence if, given any > 0, there exists N such that
%(xn , xm ) < for all n, m > N .
Lemma 6.2. Every convergent sequence is a Ca
1
Chapter 10: Compact Metric Spaces
10.1 Definition. A collection of open sets cfw_Ui : i I in X is an open
cover of Y X if Y iI Ui . A subcover of cfw_Ui : i I is a subcollection
cfw_Uj : j J for some J I that still covers Y . It is a finite subcover if
1.4. DISTRIBUTION FUNCTIONS
9
1.4 Distribution Functions
Definition 1.8. The probability of the event (X x) expressed as a function of
x R:
FX (x) = PX (X x)
is called the cumulative distribution function (cdf) of the rv X.
Example 1.7. The cdf of the rv
Homework 4 Solutions
Math 171, Spring 2010
Please send corrections to [email protected]
P p
P
P
26.5. Let n=1 an and n=1 bn be absolutely convergent series. Prove that the series n=1 |an bn |
converges.
P
P
P
Solution. Since n=1 |an | and n=1 |bn |
Chapter 3
Random Variables and
Measurable Functions.
3.1
Measurability
Definition 42 (Measurable function) Let f be a function from a measurable
space (, F) into the real numbers. We say that the function is measurable if
for each Borel set B B , the set