Page 164 - DMTH404_STATISTICS

Statistics

Notes  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
S for every  > 0 where S = X + X + ..... + X . Now, what is n ? S is the number of
n
n n 1 2 n n
S
successes observed in n trials and therefore n is the proportinn of successes in n trials.
n
Then the above 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.

11.3 Keywords

Chebyshev’s inequality is discussed, as an application, weak law of large numbers is derived.
Weak law of large nombers: Suppose X , X , ....., X are i.i.d. random variables with mean m and
1 2 n
2
finite variance  .
11.4 Self Assessment

1. Computation of the probabilities, even when the .................. functions are known, is
cumbersome at times.
(a) Chebyshev’s inequality (b) limiting distributions
(c) (absolutely) continuous (d) probability distribution
2. The .................. can be used for computation of the probabilities approximately.
(a) Chebyshev’s inequality (b) limiting distributions

(c) (absolutely) continuous (d) probability distribution
3. .................. is discussed, as an application, weak law of large numbers is derived.
(a) Chebyshev’s inequality (b) limiting distributions

(c) (absolutely) continuous (d) probability distribution
4. Chebyshev’s inequality also holds when the distribution of X is neither .................. nor
discrete.

(a) Chebyshev’s inequality (b) limiting distributions
(c) (absolutely) continuous (d) probability distribution

11..5 Review Questions

2
1. Suppose X is N(,  ). Then E(X) =  and Var(X) =  . Let us compute P[ |X – |  3].
2
2. Suppose X is a random variable such that P[X = 1] = 1/2 = P[X = –1]. Let us compute an
upper bound for P[|X – |> 1/2].
3. Suppose a person makes 100 check transactions during a certain period. In balancing his or
her check book transactions, suppose he or she rounds off the check entries to the nearest

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