EL6303
Midterm Exam Solutions
Spring 2011
1.
Three types of messages arrive at a message center:
“high priority,” denoted by the letter H,
“normal priority,” denoted by the letter N, and “low priority,” denoted by the letter L.
Assume that
1
.
0
)
(
=
Η
Π
,
4
.
0
)
(
=
Ν
Π
,
5
.
0
)
(
=
Λ
Π
a)
Find the probability that out of 20 messages, 5 are high priority, 10 are normal priority,
and
5 are low priority.
This is generalized bernouli trials and the answer is
5
10
5
)
5
.
0
(
)
4
.
0
(
)
1
.
0
(
!
5
!
10
!
5
!
20
b)
Suppose that messages arrive one at a time.
Find the probability that the first high
priority message arrives at the
th
k
message.
This event looks like
{
}
H
H
H
H
H
H
H
H
,
,
,
.
.
.
,
,
,
,
,
which is
1

κ
s
H
'
followed by one
H
.
Since they are independent we get
{
}
{ }
{ }
)
1
.
0
(
)
1
.
0
1
(
,
,
,
.
.
.
,
,
,
,
,
1
1



=
=
k
k
H
P
H
P
H
H
H
H
H
H
H
H
P
2.
X
is a uniform random variable on the interval (0,2).
Let
)
(
X
g
Y
=
where
≥
<
≤
=
1
,
1
0
,
)
(
2
x
x
x
x
x
g
(a)
Find and sketch
)
(
y
F
Y
.
For
1
0
<
<
ψ
{
}
{
}
{
}
)
(
)
(
2
y
F
y
X
P
y
X
P
y
Y
P
y
F
X
Y
=
£
=
£
=
£
=
But it is easy to find that
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<
<
=
otherhwise
x
for
x
x
F
X
;
0
1
0
;
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 Spring '11
 voltz
 Probability theory, 2 K, one second, λ, high priority, 5 1 1 2 l

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