RDP 2015-10: The Life of Australian Banknotes
Equation
E
(
t
)
=
∫
t
l
(
t
)
d
t
=
∫
0
∞
t
e
x
′
β
/
k
k
λ
(
e
x
′
β
/
k
t
λ
)
k
−
1
(
e
−
(
e
x
′
β
/
k
t
λ
)
k
)
d
t
=
∫
0
∞
k
(
e
x
′
β
/
k
t
λ
)
k
(
e
−
(
e
x
′
β
/
k
t
λ
)
k
)
d
t
=
∫
0
∞
k
u
(
e
−
u
)
λ
k
e
x
′
β
/
k
u
1
k
−
1
d
u
where
u
=
(
e
x
′
β
/
k
t
λ
)
k
and
d
u
d
t
=
k
t
k
−
1
λ
k
(
e
x
′
β
/
k
)
k
=
λ
e
x
′
β
/
k
∫
0
∞
u
1
k
(
e
−
u
)
d
u
=
λ
e
x
′
β
/
k
Γ
(
1
+
1
k
)
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