Page 162 - DMTH404_STATISTICS

Statistics

Notes Var(S )
P  |S   0 | 5)  100
100
25
100
=
12 25

1
= .
3
Here are some exercises for you.
The above examples and exercises must have given you enough practise to apply Chebyshev’s
inequality. Now we shall use this inequality to establish an important result.
Suppose X , X , ....., X are independent and identically distributed random variables having
1 2 n
2
mean  and variance  . We define
1 n
X n   X i
n i 1

 2
Then X has mean  and variance . Hence, by the Chebyshev’s inequality, we get
n
n
 2


P |X n   |  
  n 2
 2
for any  > 0. If n  0, then 2  0 and therefore
n
P  |X   |    0.
n

In other words, as n grows large, the probability that X differs from  by more than any given
n
positive number E, becomes small. An alternate way of stating this result is as follows :
For any  > 0, given any positive number , we a n choose sufficiently large n such that

P  |X   |    
n
This result is known as the weak law of large numbers. We now state it as a theorem.
Theorem 2 (Weak law of large nombers) : Suppose X , X , ....., X are i.i.d. random variables with
1 2 n
mean  and finite variance  .
2
Let

1 n
X n   X . i
n i 1

Then



P |X n   |   0 as n   .
 
for any E > 0.

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