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Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2010
Homework #5
Due: 11/8/10, in class
is restricted to ¨¦¤¢
©§¥£ , where , of course, and ¨ is 4
¦ 9" B
TDACAB
E
4
4
¦T9S"RB E CA@Q¨ E
9
9¢
¤
4
E 9" B
FDACA@¢
9 ¨ ¤ ¤¨ ¦ ¤ ¢ ©§¥£¡ 1. Consider the probability space
deﬁned by:
'( ¤¨4
§82753 6 4"
6"
P3 1 1
¤¨
V2U3 6"
G
1
2(0
H
I¨ 4 26 E ) % ¨¨"¤ &¡$#!£¡ 1 where is a constant.
1 (a) Find c.
E
G X is more than away from .
4 , the probability that the outcome E ¨E `
¥W E
G 6 W YX
4 (b) Find Deﬁne the following random variables:
¤
¥¢ pi¦ 9 X 9
q©hgf@¢ ¤
q¦ ¦ 9 X B pi
Tgg@q©¦ ¤
q¦ uiw 9 X B ui
vxAgfv©¦ ¤
¥¢ ¦ 9 X B ui
Tgfvxw .
r (0 ) %¨X
scb r , the probability density function of a a '( ¤
¥¢ ui¦ 9 X 9
v©Tgft¢ d %¨X
eccb ¨
b ¨ ¤
V¡b a
qy (c) Find (d) Find
, the joint probability density function of and . (Three important notes: (1)
Your solution to (c) does not really help you with this part, except as a check; (2) If you have a
hard time writing the expression for the joint pdf, giving a different probabilistic description of
and is ﬁne; (3) Note that you can do (e) without doing this part.)
Py
r .
have joint probability density function d ¤
¥¢ and ¦ 9 W W 9 ¤¦ 9 9 ¢ ¤4 4
T¥I ¡qhf@F a % ¨ ¤
&V¡b 1
2. The random variables r r
a a (e) Find otherwise
Py
(a) Find the value of .
.
h% r ¨
a 6¦
7TB r 1 (b) Write an expression (no need to evaluate) for
¨W
V b a (c) Find
, the conditional probability density function of given
. For your limits
(which you should not forget), use
and then bound in terms of .
a
W such that ¨ ¦g ¢ B W j
©m¥fVlk6 j
¦ 99¦
hfed6 . Given this information, ﬁnd the hg ¢
i¥f%
r
D y
(d) Somebody tells you that
is maximized. r have the following joint probability description: %
C p¤ ¢¤
#¥p 6 ¡ g % W
H
@ W
d
¢ otherwise 6 %¨
sVf% and g¦¤ ¢¤
q§¥q¦ a r
¤ % 3. Random variables a C 1 (a) Find the value of the constant .
1 r
a r
a ¨ .
.
and a Ib W £
§ s
¢
4 ¨ f%
¢ a b j
W £
§ s
¢ ¨p
x6 , % % a Ib W . %
a
r 6 ©¨ p h g
3Vq ¥¢ else
r is uniformly distributed on
d %¨
skb , the random variable ¡¥¢
¤ ¢ y
%
a We also know that, given is given by: ¤
¨ 4 a 4. The marginal density function of p99
Aft¢ r ¥£
¦¤ c
¢
4 independent?
a r and . ¨
xp B
a (h) Are for all .) .
¨
(g) Sketch r (f) Find % (e) Find ¨ (c) Find the marginal probability description of
(d) Find (i.e. (b) Find the marginal probability description of . Note: Being very careful with your limits is the key to this problem.
¨ W
b .
y (c) Find the marginal probability density function ¨ b
P y
(b) Find the joint probability density function ¨ ¤
V¡b s
y
(a) Find .
. 5. Jointly Gaussian random variables: ¢ , and
,
and ,
r %
r4 a a a ¢ %
r
C ¢ %
a ¨xwA`
C r rw 6 a f%
p E %
P8
a
u r % 4
a r
and
be jointly Gaussian random variables. Suppose you know that
, and that and are independent. By doing measurements, you ﬁnd that
and that
. Find possible values for the pair ( ,
), the variances of
.
respectively, and (do not forget this part) the correlation coefﬁcient u
f% 4 a , 4 . , %
C
C r 4
P!
4
¢
T% r a h % 4 (b) Let and
be jointly Gaussian random variables. Let
, and
. Deﬁne
. Find u (a) Let r H4 a
C
6. The money (in thousands of dollars) made from investing in stocks “Ystock” and “Zstock” are modeled as the random variables and , respectively. Assume and are independent with respective
probability density functions
and
as shown below:
r ¨¢
£ ¨¢
8£ ¨ ¡y r
b
y ¨ b ¡y ¥
u
qy ¥
u
w w
p p
¦ ¢¤
¦
G E E
4 ¦ G ¤
¦
G E
4 E
G (a) You want to make as much money as possible, of course. Which stock would you buy?
(b) Suppose you decide to buy “Ystock” and your friend decides to buy “Zstock”. What is the probability that you make more money? (In other words, ﬁnd
).
¨ ` r ...
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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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