Download Theorem The distribution of max{X 1, X2, ..., Xn}, where Xi ∼ power

Theorem The distribution of max{X1 , X2 , . . . , Xn }, where Xi ∼ power(α, βi ), i = 1, 2, . . . , n
are mutually independent random variables, has the power distribution.
Proof The power distribution has probability density function
f (x) =
βxβ−1
αβ
0<x<α
β
0<x<α
and cumulative distribution function
F (x) =
x
α
for α > 0 and β > 0. Let Y = max{X1 , X2 , . . . , Xn }. The cumulative distribution function
of Y is
FY (y) = P (max{X1 , X2 , . . . , Xn } ≤ y)
= P (X1 ≤ y, X2 ≤ y, . . . , Xn ≤ y)
= P (X1 ≤ y) P (X2 ≤ y) . . . P (Xn ≤ y)
βn
β1 β2
y
y
y
...
=
α
α
α
P n βi
i=1
y
=
0 < y < α.
α
This is recognized as the cumulative distribution function of a power(α,
variable.
1
Pn
i=1
βi ) random