University of Illinois
Spring 2010
ECE 313: Problem Set 7: Solutions
Decision Making; Independent Events; System Reliability
1. [
Mutually exclusive events
] Let
C
=
A
!
B
( )
c
denote the event that neither A nor B occurs
on a trial of the experiment, and notice that one of the three events A, B, and C always occurs on
a trial. Then, A occurs before B does if for n = 1; 2; 3, … , C occurs on the 1st, 2nd, … , (n  1)
th trials and A occurs on the nth trial. By independence of the trials,
P A before B
{ } =
P A
( ) +
P C
( )
P A
( ) +
P C
( )
!
"
#
$
2
P A
( ) +
...
=
P A
( )
1
%
P C
( )
=
P A
( )
P A
( ) +
P B
( )
Thus,
P Bbefore A
{ } =
1
!
P A before B
{ } =
P B
( )
P A
( ) +
P B
( )
.
The way to think about this is we can ignore all trials on which C occurs. On the very first trial
on which one of A and B occurs, what are the chances that A occurs?
Thus,
P A A
!
B
( ) =
P A
( )
P A
!
B
( )
=
P A
( )
P A
( ) +
P B
( )
.
2. [
Detection problem with geometric distribution vs. Poisson distribution
]
(a)
Mean
Variance
Standard deviation
H
0
λ
= 10
λ
= 10
3.162
H
1
1/p = 10
(1 p)/p
2
= 90
9.487
Note: The means are the same but the distribution under H
1
is more spread out (e.g. its standard
deviation is three times larger) than the distribution under H
0
. Moreover, on one hand, the
geometric pmf with p = 10 is slowly decreasing, with values p(1) = 0.1, p(10) = 0.0387, p(20) =
0.0135; and p(30) = 0.00471, for example. On the other hand, the Poisson pmf for
λ
= 10
has a
peak value at k = 10 and falls off rapidly away from k = 10.
(b) The ML rule is to decide H1 is true if
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 Spring '08
 Sarwate
 Probability theory, 1 %, University of Illinois, Poisson PMF

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