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Sums of Independent Random
Variables
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4. Approximations
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Theorem 4.1 (Chebyshev
Inequality)
LetXbe random variable with mean and variance 2.
Then for any and let > 0
2
2.P X
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Proof
2
2
2
2
x
x e
V X x f x dx
x f x dx
f x dx
P X
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Theorem 4.2 (Law of Large
Numbers)
LetX1,X2, ... ,Xnbe an independent and identically
distributed random variables with mean and variance
2. Let Sn=X1+X2++Xn. Then for any real number
> 0 we have
lim 0 .n
n
SP
n
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Proof
12
2
2 21
2
2 0
Note that
1 1
1 1 .
By Chebyshev inequality
lim 0 .
nn
n
in
nn
i
n n
x
SE E S
n n n
SVar Var S
n n n n
S SnP P
n n
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Example 4.1
Let Ebe an event and define independent randomvariablesXisuch thatXiis 1 if the event Eoccurs
andXiis 0 otherwise. Letp= P(E).
Then E(Xi) =pand
This gives a mathematical justification to the
relative frequency interpretation of probability.
lim 0.nn
SP p
n
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Convergence in Distribution
LetX1,X2, ... be a sequence of random variablesand let Fibe the distribution ofXifor each i. LetX
be a random variable with distribution F. We say
thatXnconverges in distribution toXif
lim for all .nn
F x F x x
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Theorem 4.3
LetX1,X2, ... be a sequence of random variablesand let ibe the mgf ofXi for each i. LetXbe a
random variable with mgf . ThenXnconverges in
distribution toXif
for all tin some neighborhood of 0.
lim nn
t t
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Poisson Approximation to the
Binomial
LetX1,X2, ... be independent Binomial randomvariables with parameters nandp. Suppose that
limnnp= . ThenXnconverges in distribution to a
Poisson random variable with mean .
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1
1
11
1Now, lim 1 and lim .
Hence,
lim .
t
t
nt
nt
n
nte
n n
e
nn
np et pe q
n
ee np
n
t e
Proof
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Example 4.2
An airline finds that 4 percent of the passengersthat make reservations on a particular flight will not
show up. Consequently, their policy is to sell 100
reserved seats on a plane that has only 98 seats.
Find the probability that every person who showsup for the flight will find a seat available.
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Solution
LetXbe the number of passengers who do notshow up. ThenXis binomial with parameters n=
100 andp= .04 . We want P(X 2). Using Poisson
approximation
4 42 1 4 .908.P X e e
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Theorem 4.4
LetX1,X2, ... ,Xnbe independent and identicallydistributed Normal random variables with mean
and variance 2. Define
Then has a standard Normal
distribution.
1 2
.
n
n
X X X
X n
n nZ n X
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Proof
2
1
2
1 1
Let . We have shown previously that
has a standard normal distribution.
1 .
n i
i i
i
nn
n i
i
n nt n
Z Y
i i
Y X
Y
XZ n Yn
tt e
n
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Theorem 6.5 (Central Limit Theorem)
LetX1,X2, ... ,Xnbe independent and identicallydistributed random variables with mean and
variance 2. Define
Then converges in distribution to
a standard Normal distribution.
1 2 .nnX X XX
n
n nZ n X
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Proof
2
2
Let . Then
1 0 0 0
2with lim 0.
But 0 1, 0 0 and 0 1. Hence
1 .2
i
i i
i
i i
Y n
nn
Y Y
Y n
Y X
t t tR
n n nnR
t tR
nn
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2
2
1 1
22
Thus
12
2 1
as .
n i
n n
Z Y n
i i
ntn
t tt R
nn
t nRe
n
n
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