Page 279 - DMTH404_STATISTICS

Unit 20: Control Limit Theorem

As usual we suggest that you go back to the beginning of the unit and see if you have Notes
achieved the objectives. We have given our solutions to the exercises in the unit in the last
section. Please go through them too. With this we have come to the end of this block.
 The Central Limit Theorem (CLT) is one of the most important and useful results in
probability theory. We have already seen that the sum of a finite number of independent
normal random variables is normally distributed. However the sum of a finite number of
independent non-normal random variables need not be normally distributed. Even then,
according to the central limit theorem, the sum of a large number of independent random
variables has a distribution that is approximately normal under general conditions. The
CLT provides a simple method of computing the probabilities for the sum of independent
random variables approximately. This theorem also suggests the reasoning behind why
most of the data observed in practice leads to bell-shaped curves.

20.3 Keywords

Binomial distribution with parameters n and p is shown to be approximable by a Poisson
distribution whenever n is large and p is such that np is a constant A z 0.
Central Limit Theorem (CLT): The Central Limit Theorem (CLT) is one of the most important
and useful results in probability theory.

20.4 Self Assessment

1. .................. with parameters n and p is shown to be approximable by a Poisson distribution
whenever n is large and p is such that np is a constant A z 0.
2. An important special case of this result is that binomial distribution can be approximated
by an appropriate .................. for large samples.
3. The .................. is one of the most important and useful results in probability theory.
4. The CLT provides a simple method of computing the probabilities for the sum of ..................
approximately.

20.5 Review Questions

1. If X is binomial with n = 100 and p = 1/2, find an approximation for P[X = 50].
2. Suppose X is binomial with parameters n and p = 0.55. Determine the smallest n for which

 X 1 
P   0.95
 
 n 2 
approximately.

3. If 10 fair dice are rolled, find the approximate probability that the sum of the numbers
observed is between 30 and 40.
4. Suppose X is binomial with n = 100 and p = 0.1. Find the approximate value of P(12  X 
14) using
(a) the normal approximation
(b) the poisson approximation, and

(c) the binomial distribution.

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