7/27/2019 For math 181

1/18

Sums of Independent Random

Variables

7/27/2019 For math 181

2/18

4. Approximations

7/27/2019 For math 181

3/18

Theorem 4.1 (Chebyshev

Inequality)

LetXbe random variable with mean and variance 2.

Then for any and let > 0

2

2.P X

7/27/2019 For math 181

4/18

Proof

2

2

2

2

x

x e

V X x f x dx

x f x dx

f x dx

P X

7/27/2019 For math 181

5/18

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

7/27/2019 For math 181

6/18

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

7/27/2019 For math 181

7/18

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

7/27/2019 For math 181

8/18

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

7/27/2019 For math 181

9/18

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

7/27/2019 For math 181

10/18

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 .

7/27/2019 For math 181

11/18

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

7/27/2019 For math 181

12/18

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.

7/27/2019 For math 181

13/18

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

7/27/2019 For math 181

14/18

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

7/27/2019 For math 181

15/18

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

7/27/2019 For math 181

16/18

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

7/27/2019 For math 181

17/18

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

7/27/2019 For math 181

18/18

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