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Rensselaer
Electrical, Computer, and Systems Engineering Department
ECSE4500 PEA
Exam 1 Solutions
March 19, 2010
Closed book  One sheet of notes allowed.
Please show work for full credit. Only hand in your answer booklet
.
Good luck!
1. (30%) Often we are faced with determining the
more likely
of two alternatives. In such a
case we are given two probability measures for a single sample space and
f
eld of events, i.e.
(
Ω
,
F
,P
1
)
and
(
Ω
,
F
2
)
, and we are asked to determine the probability of an observed event
E
in both cases.
The
more likely
alternative is said to be the one which gives the higher
probability of
E
.
Consider that two coins are in a box, one is ’fair’ with
P
1
[
{
H
}
]=0
.
5
and one is ’biased’ with
P
2
[
{
H
}
]=
p
. Without looking, we draw one coin from the box and then
F
ip this single coin
10 times. We only consider the repeated coin
F
ipsasourexper
imentandsothesamp
lespace
Ω
=
{
all tencharacter strings of H and T
}
.
We observe the event
E
=
{
a total of four Hs
and six Ts
}
.
(a) (20%) What are the two probabilities of the observed event
E,
i.e.
P
1
[
E
]
and
P
2
[
E
]
?
Ans:
This is Bernoulli trials. Thus
P
1
[
E
¡
10
4
¢¡
1
2
¢
10
and
P
2
[
E
¡
10
4
¢
p
4
(1
−
p
)
6
.
(b) (10%) Determine the
likelihood ratio
L
,
P
1
[
E
]
/P
2
[
E
]
as a function of
p.
(When
L>
1
we say that the fair coin is more likely. This test is called a
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