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Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2010
Homework #6
Due: 11/12/10, in class
1. Consider an experiment where I draw a piece of fruit from a shopping bag. The bag contains an
Apple, Banana, Lime, Pear, and Orange. Since I draw them equally likely, I deﬁne the probability
space
as:
¨£ ¥£ ¡
©§¦¤¢ 3 ' 1
20)(&%$
' ¨
£ power set ¥5
64£ 1
' !
#"
£
¥
¡
Apple, Banana, Lime, Pear, Orange The fruit rots at different rates depending on the type. Using advanced science, I calculate how
“edible” a fruit is on day to arrive at the sequence of random variables
deﬁned by:
@8
A9 7
TD £
£
TD
£
TD
£ e
f8 I R
VW
7 £
TD H
HH R
VU 7
TD GHH R
SQ
7 HH RX
7 D @
0FECB8 HH
HP 7 R
V3 . converge? If so, to what and in what ways?
h
¦£
5
D
D
#FEC§8
D @
x
@
v£r
¥ wQ ut sqh
c
dE b`
FaY
@8
g9 (b) Does the probability density function of (a) Find Apple
Banana
Lime
Pear
Orange ¨£ ¥£ h
©§¦pi 2. An experiment is deﬁned by the probability space
, where
,
is the Borel algebra restricted to
, and
is deﬁned by
. Let
.
Determine whether or not the sequence
converges (and to what) for each of the following cases.
Do the parts in order and be sure to justify each answer.
)26¦
(¦&¨
£ y &¨ v£
wQ ut r @8
#9 (a) In distribution.
(b) In probability.
(c) In quadratic mean.
(d) Almost surely.
for which:
Q
Q @8
§9 3. Consider the sequence of random variables t
0gBE&¨
@ 8 Q 7
0 Q @ 8 gBE&¨ 7
@8
§9
£ W £ U £ Q 7 (b) In probability. (a) In distribution. t
8
for
. Determine whether or not the sequence
converges to
following cases. Do the parts in order and be sure to justify each answer. for each of the (c) In quadratic mean.
¨£ ¥£ h
©§¦pi (d) Because the mapping from
is not given explicitly, it is not possible to determine whether
the sequence converges almost surely to
. It could be true or not:
t
8
@8
&9 such that:
Q
@ t
0
Q and deﬁnition of ¨£ ¥£ h
©§¦pi Give an 8 E&¨ 7
Q
0 Q @ 8 gBE&¨ 7 such that: @8
&9 t
0gBE&¨
@ 8 Q Q7 and deﬁnition of . Q ¨©§¦pi £ ¥£ h
@8
g9 Give an does converge almost surely to t
8
for which
0 Q @ 8 gBE&¨ 7
t
8
does not converge almost surely to converges in mean square to but does not converge with probability
, and the deﬁnition of the random variables
.)
£
¤ (restricted to , of course), and e ¡ , v£
wQ ut r 8 v£
wQ ut r §
¨D
@
¡ ¥ @ §
©D e e ¥ £
ut ¥
¦t @ §
©D @ @ ¨£ ¥£ h
©#¦pi ¡£ ¡
¢¤¢ ¨
©£ be given by
, let D FEC@ @8
&9 ¡ 5 D 5. Let the probability space
. For @8
g9 4. Give an example where
one. (Be sure to give . 8 for which
u i ¦
£ (¦#¨ GHH
HH
HH
HH
HH £@
HH
HH
HH
HH £ @
HH
HH
H I
HH FECB8
D @ e HH
@
§
©D ¥ @ HH
HH £ @
HH
HH
HH
HH
HH
Q ¥
©D ¥ @
e
d@ @
£
e
d@ HH
HH
HP Does this sequence of random variables converge? If so, to what and in what sense (almost surely, in
probability, in mean square, in distribution)? ...
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This note was uploaded on 09/16/2011 for the course ECE ECE603 taught by Professor Dennisgoeckel during the Fall '10 term at UMass (Amherst).
 Fall '10
 DennisGoeckel

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