Page 276 - DMTH404_STATISTICS

Statistics

Notes Let us now state the main theorem.

Theorem 3 (Central Limit Theorem) : Let X , X , ........ be an infinite sequence of independent and
1 2
2
identically distributed random variables with mean p and finite variance  . Then, for any real
x,

 X  ... X  n 
P 1 n   (x) as n   ...(5)
 
  n 
where (x) is the standard normal distribution function.
We have omitted the proof because proof of this result involves complex analysis and other
concepts which are beyond the scope of this course. Let us try to understand the above statement

more clearly. Let S = X + X +....+ X . Then we know that P  S  n  x represents the
n
n 1 2 n  
  n 
S  n
distribution of the random variable n . Then the theorem says that the distribution of
 n
S  n is approximately a standard normal distribution for sufficiently large n. Therefore the
n
 n
2
distribution of S will be approximately normal with mean n and variance n . In other words
n
the theorem asserts that if X + X ... + X are i.i.d.r. v’ s of any kind (discrete or continuous) with
1 2 n
finite variances, the  S = X + X + .... + X will approximately be a normal distribution for
n 1 2 n
sufficiently large n. The importance of the theorem lies in this fact. This theorem has got many
applications. An important application is to a sequence of Bernoulli random variables.
Normal Approximation to the Binomial Distribution

Let X , i  1 be a sequence of i.i.d. random variables such that
i
P[X = 1] = p, P[X = 0] = 1 – p
i i
where 0 < p < l.
Observe that S = X + ...... + X has the binomial distribution with parameters n qnd p. You can
n 1 n
check that E(X ) = p and Var (X) = p (1-p) for any i which is finite and positive. An application of
i i
the central limit theorem gives the following result:
For every real x,

 Sn np 

x 
P    f(x) as n   .

  n [p(1 p)  
In other words, for large n
P[ S   np x np(1 p)   (x) ...(6)


 n 
where  denotes that the quantities on both sides are approximately equal to each other.
An alternate way of interrupting the above approximation that a binomial distribution tends
to be close to a normal distribution for large n. Let us explain this in mote detail.

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