Page 163 - DMTH404_STATISTICS

Unit 11: Chebyshev’s Inequality

The above theorem is true even when the variance is infinite but the mean p is finite However Notes
this result does not follow as an application of the Chebyshev’s inequality in this general set up.
The proof in the general case is beyond the scope of this course.
We make a remark here.
Remark 2 : The above theorem only says that the probability that the value of the difference
|X n  X| exceeds any fixed number , gets smaller and smaller for successively large values of n.
The theorem does not say anything about the limiting case of the actual difference. In fact there
is another strong result which talks about the limiting case of the actual values of the differences.
This is the reason why Theorem 2 is called ‘weak law’. We hove not included the stronger result
here since it is beyond the level of this course.
Let us see an example.

Example 4: Suppose a random experiment has two possihle outcomes called success (S)
and Failure (F). Let p he the probability of a success. Suppose the experiment is repeated
independently n times. Let X take the value 1 or 0 according as the outcome in the i-th trial of the
i
experiments is success or a failure. Let us apply Theorem 2 to the set   n j 1 .
X
i

We first note that
P[X = 1] = p and P[X = 0] = 1 – p = q,
i i
for 1  i  n. Also you can check that E(X ) = p ond var (X ) = p q for i = 1, ..... n.
i i
Since the mean and the variance are finite, we can apply the weak law of large numbers for the
sequence {X : 1  i  n}. Then we have
1
 S 
P n  p    0 as b  
 
 n 
S
Sn for every  > 0 where S = X + X + ..... + X . Now, what is n ? S is the number of successes
n 1 2 n n
n
S
observed in n trials and therefore n is the proportinn of successes in n trials. Then the above
n
result says that as the number of trials increases, the proportion of successes tends stabilize to
the probability of a success. Of course, one of the basic assumptions behind this interpretation is
that the random experiment can be repeated.
In the next section we shall discuss another limit theorem which gives an approximation to the
binomial distrihution.

11.2 Summary

2
 Suppose X is a random variable with mean  and finite variance  . Then for every  > 0.
 The above theorem only says that the probability that the value of the difference |X  X|
n
exceeds any fixed number , gets smaller and smaller for successively large values of n.
The theorem does not say anything about the limiting case of the actual difference. In fact
there is another strong result which talks about the limiting case of the actual values of the
differences. This is the reason why Theorem 2 is called ‘weak law’. We hove not included
the stronger result here since it is beyond the level of this course.

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