•1
•
Example:
Suppose, we have particles emitted
in t seconds according to a Poisson law with
rate
Assume also that all particles have
energies
that are distributed
according to the Maxwell distribution with
mean
and variance
•
Let S denote the total energy emitted in t
seconds. Clearly, it is a random variable. Let
us , for instance, compute its mean and
variance. We have
.
t
λ
},
{
k
X
2
3
T
.
2
3
2
T
t
N
t
EN
=
=
)
var(
,
Then
.
2
3
t
T
EX
EN
ES
⋅
=
⋅
=
2
)
)(
var(
)
var(
)
var(
EX
N
X
EN
S
+
⋅
=
4
15
4
9
2
3
2
2
2
t
T
T
t
T
t
=
⋅
+
⋅
=
±
Weak Law of Large Numbers
•
Here we begin to study the behavior of
random variables
and
as n goes to
infinity.
•
It is important to estimate unknown
parameters of the distribution from
which you take samples.
•
The weak law of large numbers(WLLN)
says that for sufficiently large n random
variables:
with a high probability
takes values close to the mean of the
distribution from which you sample.
n
S
n
M
n
M
Theorem 3
with finite mean m and variance
.
Then for any fixed
•
Proof:
Applying Chebyshev’s inequality,
we obtain
•
Therefore,
as
+∞
=
1
}
{
i
i
X
2
σ
0
>
ε
1
)

(
lim
=
<
−
+∞
→
m
M
P
n
n
2
2
)

(
n
m
M
P
n
≤
≥
−
1
1
)

(
2
2
→
−
≥
<
−
n
m
M
P
n
.
+∞
→
n
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View Full Document•2
•
Example:
Let
be an IIDsequence
with EX=m and
Let
Does the WLLN hold for
since we know that the WLLN holds for
+∞
}
{
i
X
.
)
var(
2
σ
=
X
.
,
3
2
N
n
X
Y
n
n
∈
+
=
?
}
{
1
+∞
=
i
n
Y
{}
1
2
1
3
2
3
2
1
1
1
1
1
∑
∑
∑
=
+∞
→
=
+∞
→
=
+∞
→
=
<
−
=
=
<
−
−
+
=
=
<
−
n
i
i
n
n
i
i
n
n
i
i
n
m
X
n
P
m
X
n
P
Y
E
Y
n
P
)

(
lim
)

(
lim
)

(
lim
ε
+∞
=
1
}
{
i
i
X
±
Strong Law of Large Numbers
•
We know that
is an IID– sequence ,
then the WLLN holds. Here we point out
that an even stronger result takes place.
Theorem 4
Consider a sequence of IID
random variables
. Let
E{X}=m
and
.
Then
+∞
=
1
}
{
i
i
X
+∞
=
1
}
{
i
i
X
2
)
var(
=
X
.
1
)
lim
(
=
=
∞
→
m
M
P
n
n
±
Practical Significance of SLLN
•
The SLLN is the basis for estimating
different quantities in real life.
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 Fall '08
 Krim
 Probability theory, Convergence, lim P

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