Sean las variables aleatorias
X
1
,
X
2
,
.
.
.
,
X
n
{\displaystyle X_{1},X_{2},...,X_{n}}
función de distribución
F
X
(
x
)
{\displaystyle F_{X}(x)}
función de densidad
f
X
(
x
)
{\displaystyle f_{X}(x)}
Y
{\displaystyle Y}
Y
=
m
a
x
(
X
1
,
X
2
,
.
.
.
,
X
n
)
{\displaystyle Y=max(X_{1},X_{2},...,X_{n})}
F
Y
(
y
)
=
[
F
X
(
y
)
]
n
{\displaystyle F_{Y}(y)=[F_{X}(y)]^{n}}
f
Y
(
y
)
=
n
[
F
X
(
y
)
]
n
−
1
f
X
(
y
)
{\displaystyle f_{Y}(y)=n[F_{X}(y)]^{n-1}f_{X}(y)}
Supongamos que
G
(
y
)
{\displaystyle G(y)}
G
(
y
)
=
P
(
Y
≤
y
)
=
P
(
m
a
x
(
X
1
,
.
.
.
,
X
n
)
≤
y
)
{\displaystyle G(y)=P(Y\leq y)=P(max(X_{1},...,X_{n})\leq y)}
Como
X
i
≤
m
a
x
(
X
1
,
.
.
.
,
X
n
)
{\displaystyle X_{i}\leq max(X_{1},...,X_{n})}
i
=
1
,
2
,
.
.
.
,
n
{\displaystyle i=1,2,...,n}
m
a
x
(
X
1
,
.
.
.
,
X
n
)
≤
y
{\displaystyle max(X_{1},...,X_{n})\leq y}
(
X
1
≤
y
,
X
2
≤
y
,
.
.
.
,
X
n
≤
y
)
{\displaystyle (X_{1}\leq y,X_{2}\leq y,...,X_{n}\leq y)}
(
X
1
,
.
.
.
,
X
n
)
{\displaystyle (X_{1},...,X_{n})}
y
{\displaystyle y}
y
{\displaystyle y}
X
i
{\displaystyle X_{i}}
y
{\displaystyle y}
G
(
y
)
=
P
(
Y
≤
y
)
{\displaystyle G(y)=P(Y\leq y)}
=
P
(
m
a
x
(
X
1
,
.
.
.
,
X
n
)
≤
y
)
{\displaystyle =P(max(X_{1},...,X_{n})\leq y)}
=
P
(
X
1
≤
y
,
X
2
≤
y
,
.
.
.
,
X
n
≤
y
)
{\displaystyle =P(X_{1}\leq y,X_{2}\leq y,...,X_{n}\leq y)}
=
P
(
X
1
≤
y
)
P
(
X
2
≤
y
)
.
.
.
P
(
X
n
≤
y
)
{\displaystyle =P(X_{1}\leq y)P(X_{2}\leq y)...P(X_{n}\leq y)}
=
[
P
(
X
1
≤
y
)
]
n
{\displaystyle =[P(X_{1}\leq y)]^{n}}
=
[
F
(
y
)
]
n
{\displaystyle =[F(y)]^{n}}
Del mismo modo, la función de densidad de Y sería:
g
(
y
)
=
G
(
y
)
′
=
(
[
F
(
y
)
]
n
)
′
=
n
[
F
(
y
)
]
n
−
1
f
(
y
)
{\displaystyle g(y)=G(y)'=([F(y)]^{n})'=n[F(y)]^{n-1}f(y)}
![{\displaystyle F_{Y}(y)=[F_{X}(y)]^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b6f6c45465d8aa63057b44aa74808eae5b60b50)
![{\displaystyle f_{Y}(y)=n[F_{X}(y)]^{n-1}f_{X}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec538e09a02a0f9b6a16e6f83e673ae5218cd75)
![{\displaystyle =[P(X_{1}\leq y)]^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10b7a1b7870910f742cca397d8fe3e5b4aacb15a)
![{\displaystyle =[F(y)]^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b96def2603e11c5fdcd7eecd9fbf40133816f0a)
![{\displaystyle g(y)=G(y)'=([F(y)]^{n})'=n[F(y)]^{n-1}f(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f59f5bc1f1e1e6164a77f3851d761d5ffc015a7)
Datos: Q5808666