for some set
A
and its complement
A
C
in the fourth
equality.
Furthermore, there is only one pair of
(
X
1
; X
2
)
whose sum is greater than 1.9734,
i.e. only when
X
1
=
X
2
= 1
:
Lastly, since
X
1
and
X
2
are independent,
P
(
X
1
= 1
; X
2
= 1) =
P
(
X
1
= 1)
³
P
(
X
2
= 1) =
1
3
³
1
3
=
1
9
:
(c) Similarily, when
n
= 3
;
P
(
Z
3
µ
0
:
924)
=
P
°
1
p
3
3
P
i
=1
(
X
i
´
°
)
µ
0
:
924
±
=
P
°
1
p
3
°
X
1
´
1
3
+
X
2
´
1
3
+
X
3
´
1
3
±
µ
0
:
924
±
=
P
(
X
1
+
X
2
+
X
3
µ
2
:
6004)
=
1
´
P
(
X
1
+
X
2
+
X
3
>
2
:
6004)
=
1
´
P
(
X
1
= 1
; X
2
= 1
; X
3
= 1)
=
26
27
·
0
:
963
(d) So far, we have calculated the probability of
P
(
Z
n
µ
0
:
924)
for
n
= 2
in (b) and
n
= 3
in (c).
Now, we calculate the probability of
P
(
Z
n
µ
0
:
924)
for large
n
using the central limit
5
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theorem. As
n
! 1
;
the central limit theorem states,
p
n
±
²
X
n
´
°
³
!
d
N
(0
;
1)
where
X
n
=
1
n
P
n
i
=1
X
i
is the sample average of independent random variables with
°
=
E
(
X
i
)
and
±
2
=
V ar
(
X
i
)
for
i
= 1
;
2
; :::; n:
Note that "
!
d
" denotes the "covergence in distribution", i.e.
the distribution of
p
n
°
²
X
n
´
°
³
becomes arbitrarily well approximated by the standard normal
distribution
N
(0
;
1)
as
n
! 1
:
In order to apply the central limit theorem for the given random variable
Z
n
;
dividing
Z
n
by
the standard deviation
±
yields,
Z
n
±
=
1
±
1
p
n
´
n
P
i
=1
(
X
i
´
°
)
µ
=
1
±
1
p
n
´
n
P
i
=1
X
i
´
n
±
°
µ
=
p
n
±
´
1
n
n
P
i
=1
X
i
´
°
µ
=
p
n
±
²
X
n
´
°
³
!
d
N
(0
;
1)
by the central limit theorem
where we °nd that
Z
n
°
!
d
Z
with
Z
~
N
(0
;
1)
:
Given this, we can calculate
P
(
Z
n
µ
0
:
924)
as follows,
P
(
Z
n
µ
0
:
924)
=
P
°
Z
n
±
µ
0
:
924
±
±
·
P
0
@
Z
n
±
µ
0
:
924
q
2
9
1
A
·
P
(
Z
µ
1
:
96)
·
0
:
975
where
P
(
Z
µ
1
:
96)
·
0
:
975
is found from the table 1 in the appendix (Given
P
(
Z
µ ´
1
:
96) =
0
:
025
from the table, we °nd
P
(
Z
µ
1
:
96) = 1
´
P
(
Z
²
1
:
96) = 1
´
0
:
025 = 0
:
975
where
P
(
Z
²
1
:
96) =
P
(
Z
µ ´
1
:
96) = 0
:
025
due to the symmetry of the standard normal distribution.
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 Winter '08
 Stohs
 Normal Distribution, Variance, Probability distribution, Probability theory, probability density function

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