STAT 410
Examples for 06/27/2008
Summer 2008
In general, if X
1
, X
2
, … , X
n
is a random sample of size
n
from a continuous
distribution with
cumulative distribution function F
(
x
) and probability density function
f
(
x
), then
F
max
X
i
(
x
)
= P
(
max
X
i
≤
x
)
= P
(
X
1
≤
x
, X
2
≤
x
, … , X
n
≤
x
)
= P
(
X
1
≤
x
)
⋅
P
(
X
2
≤
x
)
⋅
…
⋅
P
(
X
n
≤
x
)
=
(
)
(
)
n
x
F
.
f
max
X
i
(
x
)
= F
'
max
X
i
(
x
)
=
(
)
(
)
(
)
1
F
x
f
x
n
n
⋅
⋅

.
1 – F
min
X
i
(
x
)
= P
(
min
X
i
>
x
)
= P
(
X
1
>
x
, X
2
>
x
, … , X
n
>
x
)
= P
(
X
1
>
x
)
⋅
P
(
X
2
>
x
)
⋅
…
⋅
P
(
X
n
>
x
)
=
(
)
(
)
n
x
F
1

.
F
min
X
i
(
x
)
=
(
)
(
)
n
x
F
1
1


.
f
min
X
i
(
x
)
= F
'
min
X
i
(
x
)
=
(
)
(
)
(
)
1
F
1
x
f
x
n
n
⋅
⋅


.
Let Y
k
=
k
th
smallest of X
1
, X
2
, … , X
n
.
F
Y
k
(
x
)
= P
(
Y
k
≤
x
)
= P
(
k
th
smallest observation
≤
x
)
= P
(
at least
k
observations are
≤
x
)
=
(
)
(
)
(
)
(
)
°
=


±
±
²
³
´
´
µ
¶
⋅
⋅
n
k
i
i
n
i
x
x
i
n
F
1
F
.
f
Y
k
(
x
)
= F
'
Y
k
(
x
)
=
(
)
(
)
(
)
(
)
(
)
(
)
(
)
F
1
F
1
1
!
!
!
x
f
x
x
k
n
k
n
k
n
k
⋅
⋅
⋅
⋅





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1.
Let X
1
, X
2
, X
3
, X
4
be a random sample
(
i.i.d.
)
of size
n
= 4 from a probability
distribution with the p.d.f.
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 Spring '08
 AlexeiStepanov
 Probability

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