Page 278 - DMTH404_STATISTICS

Statistics

Notes and

 149  1  135 
 2 
P[S  149] =   
n (135)(.7)
 
 
= (1.59)
Hence
P[S  150] = 1 – (1.59)
n
= .0559
This shows that the probability that more than 150 first year students attend is less than 6%.
Let us now consider a different type of application of the central limit theorem.

Example 7: Suppose X , X .... is a sequence of i.i.d. random variables each N(0, l). Then
1 2
2
2
2
X ,X , ....... is a sequence of i.i.d. random variables each with X -distribution.
1
1
2
2
2
Note that E( X ) = 1 and Var ( X ) = 2 for any i. Hence by central limit theorem we get
1 1
2
2
 X  .... X  n 

P  1 n  x   (x) as n   .

 2n 
2
2
2
But S = X + ...... + X has X distribution. What we have shown just now is that if Sn has X 2
n 1 1 n n
S  n
distribution, then n has an approximate standard normal distribution for large n. In other
2n
words, for every real x,
 S  n 
P n  x   (x)
 
 2n 
for large n whenever S, has Xz -distribution.
We make a remark now.
Remark 3 : The central limit theorem is central to the distribution theory needed for statistical
inferential techniques to he developed in Block 4. You must have noted that the distribution of
individual Xi in CLT could be discrete or continuous. The only condition that is imposed is that
its variance has to be finite. In general, it is not easy to specify the size of n for a good
approximation as it depends on the underlying distribution of {X }. However, it is found in
i
practice that, in most cases, a good approximation is obtained whenever n is greater than or
equal to 30.
We will stop our discussion on limit theorem now, though we shall refer to them off and on in
the next block. Let us now do quick review of what we have covered in this unit.
20.2 Summary

 Obtain Poisson approximation to binomial;
 Discussed the central limit theorem and obtained normal approximation to binomial as
an application.

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