Proposition 17.2
If
Y
≥
0
, then for any
A
,
P
(
Y > A
)
≤
E
Y
A
.
Proof.
Let
B
=
{
Y > A
}
. Recall 1
B
is the random variable that is 1 if
ω
∈
B
and 0 otherwise. Note 1
B
≤
Y/A
. This is obvious if
ω /
∈
B
, while if
ω
∈
B
, then
Y
(
ω
)
/A >
1 = 1
B
(
ω
). We then have
P
(
Y > A
) =
P
(
B
) =
E
1
B
≤
E
(
Y/A
) =
E
Y
A
.
We now prove the WLLN.
Theorem 17.3
Suppose the
X
i
are i.i.d. and
E

X
1

and
Var
X
1
are ﬁnite.
Then for every
a >
0
,
P
±²
²
²
S
n
n

E
X
1
²
²
²
> a
³
→
0
as
n
→ ∞
.
Proof.
Recall
E
S
n
=
n
E
X
1
and by the independence, Var
S
n
=
n
Var
X
1
,
so Var (
S
n
/n
) = Var
X
1
/n
. We have
±²
²
²
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This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.
 Spring '10
 ansan

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