CEE 3040
Fall 2008
CEE 304 – Section 4 Problems
1.
Memoryless Exponential Process
:
( )
x
X
f
x
e
λ
λ

=
for
x
≥
0
( )
1
x
X
F
x
e
λ

=

for
x
> 0
By definition:
( )
[
]
X
F
x
P X
x
=
≤
[
]
1
( )
x
X
P X
x
F
x
e
λ

∴
≥
=

=
If we already waited
t
minutes, how much longer do we expect to wait?
[i.e. What is the mean of the additional waiting time
A
?]
t
t
a
X
X
A
e
e
t
F
t
a
F
t
X
P
t
X
t
a
X
P
t
X
t
a
X
P
a
F
λ
λ

+

=

+

=
≥
≥
∩
+
≥
=
≥
+
≥
=
)
(
)
(
1
)
(
1
]
[
]
[
]

[
)
(
a
A
e
a
F
λ

=
)
(
, for a > t
Therefore, the distribution of the additional waiting time A is the same as the original waiting
time T, it is just shifted over by
t
. So,
E
[
A
] =
E
[X  X
≥
t
] =
t
+ 1/
λ
. Doesn’t matter how long
you have waited, the expected additional waiting time has mean 1/
λ
.
2. An engineering student got a summer job setting off fireworks on the 4
th
of July and other
summer events. For a particular display, the bomb is shot from the tube so that after
t
seconds
it is at height:
H = (100 m/sec) t – 0.5 (10 m/sec
2
) t
2
=
500 – 5(10 – t)
2
meters
Unfortunately the timers on the bombs are not accurate. The student estimates that the timer
will yield a time T to detonation after launch that is normally distributed with mean 10 seconds
and standard deviation 1.2 seconds.
T ~ N(
μ
= 10 sec,
σ
2
= (1.2 sec)
2
)
a.)
What is the probability that the bomb explodes more than 495 meters above the ground?
]
495
)
10
(
5
500
[
]
495
[
2
≥


=
≥
t
P
H
P
= 0.5934
b.)
Above what height can you be 99% confident the bomb will explode?
99
.
0
]
[
=
≥
h
H
P
99
.
0
]
)
10
(
5
500
[
2
=
≥


⇒
h
t
P
Ans
. h = 452.26 m
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CEE 3040
Fall 2008
c.)
What is the pdf for the height at which the bomb explodes?
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 '08
 Stedinger
 Normal Distribution, Variance, Probability theory, Exponential distribution, additional waiting time

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