This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2010
Homework #4
Due: 10/15/10, in class has probability density function:
%#
&$¦ 1. An exponential random variable with parameter !%
!
" else ¨ ¦¤ ¢
©§¥£¡ Suppose that I have three lightbulbs of varying qualities: .
D
E¤ C ¨R¦
Y£XW#
B DT R ¨ I G ¦ ¤
VUSQPH§F#
B D
E¤
C 9
6( '
' 9
@( is said to be memoryless if . B is an exponential random variable with . # is an exponential random variable with A
8 4
5( hours, where 4
6( )
0(
B Deﬁnition: A random variable
for all and . hours, where is an exponential random variable with 7
8 ' The third dies out after )
1( The second dies out after hours, where 2
3 The ﬁrst dies out after I ¦ ¨R
£EI (a) Suppose that I use the ﬁrst lightbulb (which lasts a random time ) in my lamp, and it has lasted
two hours. Given such, what is the chance it lasts more than four hours?
)( )
1( (b) Is memoryless? (c) I pick a lightbulb at random (i.e. each of the three is equally likely) and put it into my lamp. Denote
the time that the lamp is lit as . What is the probability that the lamp is lit for more than 2 hours?
( (d) I pick a lightbulb at random (i.e. each of the three is equally likely) and put it into my lamp. Denote
the time that the lamp is lit as . The lamp is lit for two hours. Given such, what is the probability
that it is lit for more than four hours?
( memoryless? Explain any difference from (b). b ¨¦
3a`§¤ ¢ ¡
b .
#4
B . 4 r 7
WG
B t
uG 4 B qi
p
A (d) Find ) rp
sB qi (c) Find the probability that . !% (b) Find is given by 2 f¦f2c T
FhWged¦ T (a) Find the value of the constant . B 2. The probability density function of a random variable otherwise ¨ ¦¤ ¢
`§¥"¡ ( (e) Is , where: 3. Suppose that your goal is to maximize the proﬁt of your business. If you decide to travel to Xville,
the proﬁt (in dollars) for your business is a random variable with cumulative distribution function
as given below. If you decide to travel to Yville, the proﬁt (in dollars) for the trip is a random
variable with cumulative distribution function
as given below.
B ¨ ¥¤
¤ ¨ ¥¤
¤ ¨¦
`§¤ ¢ ¢
£ ¡ ¢
¦ ¨¦
©§¤ ¢ 1 3 1 3 2 ¤ 2 ¨4 1 ¦ 1 ¤¤ 1 (a) To which village would you travel to maximize your proﬁt? ¨
£ . Find the cumulative distribution function .
: B § 2 % # !"
&$
( ' ¨ ¦¤
©§@ is given by ¨¦
`§¤ ¢ ¡ 4 ¨ ¦¤
Y©§@
7 . Write an , where: 2 B ¤ ¤
2f ¨ ¡ ¤ ¡ C 5. The probability density function of a random variable ; that is, ( ' 0¡
1) G4 ¡ Let be a Gaussian random variable with mean and variance
expression for
for all in terms of
. ! 4 4. You have a table that gives you the value of the “Goeckel Function” for all %#
&h¦ ¨ ¤
© (b) Let f Wf 0c
¦
7 !%
b ¨¦
3a©§¤ ¢ ¡
!4¦ otherwise (a) Find the value of the constant .
b #2
g&c
B . be deﬁned by: 2!% B B c B (where the Borel ﬁeld is restricted
2Ff
2Ff
)3 f 8
9 6 r A R ) 3 T@
R
A SQf ) 3
R@
A SQf %
R@ p 4
!% ¡ ¡ ¡
¨ C !6
q7! 4
5¤ 4
4
IP H 4 G ) EG
IP4 H DY¨ ¨ B! ¤ ¤
CE F A @
I
P H r ! ! . %
&# ! 2!% p C rp
sB qi . Find , . 6. Let the probability space
be given by
to
, of course) and, for any interval, 4 U ¨U
9a6¥¤
B Let f Find the probability density function of % (d) Let the random variable .
)3 c B (c) Find the probability that 2
F# 4 (b) Find the probability that ...
View
Full
Document
 Fall '10
 DennisGoeckel

Click to edit the document details