Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2010
Homework #3
Due: 10/8/10, in class
1. I ﬂip a fair coin twice (assume the ﬂips are independent) and record the outcome of these ﬂips in order.
For example, for “head followed by head”, the outcome is “HH”. Your job is to deﬁne a probability
model that will used by one of your coworkers to analyze the experiment. You are not aware of what
questions he/she might want to ask, so you want to generate as complete a model as possible (e.g.
you do not want to use a trivial ). Provide a probability space
for this experiment. Since
the size of the sets involved here is not that large, be explicit about how each of these three things are
deﬁned. In particular, write out all of the sets in and give the probability of each. ¨¦
©§¤ ¤¢
¥£¡ 2. A number is chosen at random from the interval [0,1]. As is the standard case, the probabilities are
deﬁned on the Borel algebra (restricted to [0,1]). Starting from ﬁrst principles (i.e. deﬁnition of
the Borel algebra, axioms of probability, etc.), answer the following three parts:
, the complement of ¤
, necessarily count
('&%
$!$ ; that is,
if and only if
. Find the probability of . must contain an and )
0!
4 31
2 ##!
" ¤
be the set of irrational numbers in
, . Is ¤
GE E E ¤ D¤ A¤ ¤ 7 "
[email protected]98PI (c) Let
for all be an arbitrary uncountable subset of (b) Let
able? algebra. Show that
(a) Let
be a subset of
that is not in the Borel
uncountable number of elements. GE E E ¤ D¤ A¤ ¤ 7 "
[email protected]9865 3. I have analyzed two independent experiments, Experiment 1 and Experiment 2, to arrive at two separate probability spaces:
and
, where: 1 ¨GA7
caXr¡ V ¦ , AE
v 1
1 ¨ b¡ S
q£Xp¦
¨b
q£¡ V ¦ is deﬁned by , ¨ G 7¡ S
cc`rXU¦ ¨g ¡ S
ihXU¦
¨g
ih¡ V ¦ ¤S Q
FTR¡
¨Q
wR¡ S ¦
1
GGA¤7¤GA7¤G7¤ b
`[email protected]`efaXefc`edc7
S
1
¨Q
wR¡ V ¦
1
V
G G s¤ A7¤ G s7¤ G A7¤ b
`xCXefefaXedc7
1 , and . is deﬁned by 1 ¨S ¦¤
U§S , and . sE
t ¤V Q
XWR¡ , uvE
¨GA7
caXr¡ S ¦
1
GA¤
[email protected]`7
S
YQ
1 , , ¨V ¦¤
9¥§@V , , , yvE
¨Gs7
cr¡ V ¦
1
VQ
G s¤ A
xCX7
1 (a) Are these valid probability spaces? Be sure to tell me all of the conditions that you checked to
arrive at your answer.
(b) My boss asks me to deﬁne a combined experiment as follows: Perform Experiment 1 and remember the result; then, perform Experiment 2 and remember the result. Now, write in your notebook
the outcome of the combined experiment as an ordered pair with the ﬁrst entry equal to the result of
Experiment 1 and and the second entry equal to the result of Experiment 2. For example, an outcome
might be “(1,4)”. Find
for the combined experiment. Use a that captures as many events
as possible, and be sure to write out explicitly at least half of the events in . ¨¦
§¤ ¤Q
YR¡ (c) Alas, the boss is ﬁckle and changes his mind. Now he asks: perform Experiment 1 and remember
the result; then, perform Experiment 2 and remember the result. Now, write in your notebook the
equal to the result of Experiment 1
outcome of the combined experiment as a random variable
times the result of Experiment 2. For example, an outcome might be “4” (which is
). Find
A
0#A . Hint: Feel free to deﬁne
¦ for the random variable
this makes it easier to represent. using the integral of a function if ¨¦
§¤ ¤ Q
R¡ Now, your buddy in the modeling department comes to you with yet another experiment description:
, where ¨ ¡ §¤ ¡
¦ ¦¤
§ 1 ¨b
q£¡ ¦ is deﬁned by ¨G7
cc`r¡ , and ¨g
ih¡ ¡ ¦
¨GD¤ A¤7
[email protected]`r¡
1
G CB[email protected]`efaBCXefc`e¤ c7
G D¤ A¤7¤GD¤ A7¤G7
b . 1 1 ¡
¦
§¤ ¢ , E
1 GD¤ A¤
[email protected]`7
1 ¨GD¤ A7
caBCXr¡ ¤¡ Q
R¡ ¡Q
¦¤ E
§@ (d) Is
a valid probability space? Be sure to tell me all of the conditions that you checked
to arrive at your answer.
¦
¡ §¤ ¡ ¨ , of course, and
¦ ¨A¤
fBH¡ is restricted to ¨g
ih¡ is V ¡ $§
Y ¥
$ ¡
V
¥$
% $ A §
$ A §
£ ¡ V
$¥ ¤¨
¤¨ V ¤¨ ¥ ¨¦
§¤
V V
V
V ¥ V
¥
§
¡ £ ¨ A¤ ¡
¤¤ fBHx¡
§
¡ © "
#!
§
¡ ¡
1 ¨¨ § ¥¡
xF¨¤ ¦x¡ be the outcome of the experiment.
¦ ¨
9(%
¨
fA
% S $ V¡ V ¡¡
¦ ¦ (b) What is Q
R¡ (a) What is ¤¡ where is a constant. Let , where 4. Consider the probability space
deﬁned by: to ﬁnd the probability that a “3” is ¨ ¡ §¤ ¡
¦ Q
¡ R¡ ¤ (e) Your boss asks you to use the description of
observed. How do you respond? ? Your answer should be a number.
? Be sure to derive everything from ﬁrst principles! (c) Since
, it can be treated as a random variable. Find and roughly sketch the cumulative
and probability density function
of .
distribution function (CDF)
&
'" ¨
q! ¡ )
21 ¨
q! ¡
4V )
0( (d) Let the random variable be deﬁned as
. Find the probability space
for
. Hint: This looks like a “function of random variable” problem, which we have not studied yet, but
you can still easily solve it from ﬁrst principles.
¨ 5¦
6§¤ ¤ £¡
¢ A 1 3 3 3 ...
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Full Document
 Fall '10
 DennisGoeckel
 Probability, Probability distribution, Probability theory, Borel

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