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Using the power series expansion of φ about the origin noted above, we have
This implies
so that
A standard lemma of analysis ensures
It is a well known property of the exponential that
so that
By the convergence theorem on characteristic functions, above, ${F}_{n}\left(t\right)\to \Phi \left(t\right)$ .
— $\square $
The theorem says that the distribution functions for sums of increasing numbers of the X _{i} converge to the normal distribution function, but it does not tell how fast. It is instructive to consider some examples, which are easily worked out with the aid of our m-functions.
Demonstration of the central limit theorem
Discrete examples
We first examine the gaussian approximation in two cases. We take the sum of five iid simple random variables in each case. The first variable has six distinct values; thesecond has only three. The discrete character of the sum is more evident in the second case. Here we use not only the gaussian approximation, but the gaussian approximationshifted one half unit (the so called continuity correction for integer-values random variables). The fit is remarkably good in either case with only five terms.
A principal tool is the m-function diidsum (sum of discrete iid random variables). It uses a designated number of iterations of mgsum.
X = [-3.2 -1.05 2.1 4.6 5.3 7.2];PX = 0.1*[2 2 1 3 1 1];EX = X*PX'
EX = 1.9900VX = dot(X.^2,PX) - EX^2
VX = 13.0904[x,px] = diidsum(X,PX,5); % Distribution for the sum of 5 iid rvF = cumsum(px); % Distribution function for the sum
stairs(x,F) % Stair step plothold on
plot(x,gaussian(5*EX,5*VX,x),'-.') % Plot of gaussian distribution function% Plotting details (see
[link] )
X = 1:3;
PX = [0.3 0.5 0.2];
EX = X*PX'EX = 1.9000
EX2 = X.^2*PX'EX2 = 4.1000
VX = EX2 - EX^2VX = 0.4900
[x,px]= diidsum(X,PX,5); % Distribution for the sum of 5 iid rv
F = cumsum(px); % Distribution function for the sumstairs(x,F) % Stair step plot
hold onplot(x,gaussian(5*EX,5*VX,x),'-.') % Plot of gaussian distribution function
plot(x,gaussian(5*EX,5*VX,x+0.5),'o') % Plot with continuity correction% Plotting details (see
[link] )
As another example, we take the sum of twenty one iid simple random variables with integer values. We examine only part of the distribution function where most of theprobability is concentrated. This effectively enlarges the x-scale, so that the nature of the approximation is more readily apparent.
X = [0 1 3 5 6];PX = 0.1*[1 2 3 2 2];EX = dot(X,PX)
EX = 3.3000VX = dot(X.^2,PX) - EX^2
VX = 4.2100[x,px] = diidsum(X,PX,21);F = cumsum(px);
FG = gaussian(21*EX,21*VX,x);stairs(40:90,F(40:90))
hold onplot(40:90,FG(40:90))
% Plotting details (see
[link] )
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