CEE 304 – Section 4 Problems (9/20/2006)
1.
Transforming Variables Example
:
( )
x
X
f
x
e
−
=
for
x
> 0,
and 0 otherwise
Develop the density function for Y = X
½
Y = g(X) = X
½
=> X = Y
2
,
y
dy
dx
2
=
Onetoone transformation: we have a onetoone relationship between X and Y, so we can
use the following formula to find the pdf of Y given the pdf of X:
( )
( )
( )
[
]
( )
[
]
dy
dx
y
x
f
dy
dx
dx
y
x
dF
dy
y
dF
y
f
X
X
Y
Y
⋅
=
=
=
2
( )
(2 )
y
Y
f
y
e
y
−
∴
=
⋅
for y > 0,
and 0 otherwise
2.
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
−
∴
≥
=
−
=
Additional lifetime (i.e. how much long it will last) = A
t
t
a
X
X
e
e
t
F
t
a
F
t
X
P
t
X
t
a
X
P
t
X
t
a
X
P
λ
λ
−
+
−
=
−
+
−
=
≥
≥
∩
+
≥
=
≥
+
≥
)
(
)
(
1
)
(
1
]
[
]
[
]

[
a
e
λ
−
=
By definition:
(
)
( )
t
X
e
t
F
t
X
P
λ
−
=
−
=
≥
1
, for
t
≥
0
Therefore, the distribution of the additional lifetime A is exactly the same as the original
distribution of waiting time T.
So,
E
[
A
] = 1/
λ
.
1
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