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Absolutely continuous examples

By use of the discrete approximation, we may get approximations to the sums of absolutely continuous random variables. The results on discrete variables indicatethat the more values the more quickly the conversion seems to occur. In our next example, we start with a random variable uniform on ( 0 , 1 ) .

Sum of three iid, uniform random variables.

Suppose X uniform ( 0 , 1 ) . Then E [ X ] = 0 . 5 and Var [ X ] = 1 / 12 .

tappr Enter matrix [a b]of x-range endpoints [0 1] Enter number of x approximation points 100Enter density as a function of t t<=1 Use row matrices X and PX as in the simple caseEX = 0.5; VX = 1/12;[z,pz] = diidsum(X,PX,3);F = cumsum(pz); FG = gaussian(3*EX,3*VX,z);length(z) ans = 298a = 1:5:296; % Plot every fifth point plot(z(a),F(a),z(a),FG(a),'o')% Plotting details (see [link] )
Figure four is a distribution graph. It is titled, distribution for the sum of three iid uniform random variables. The horizontal axis is labeled, x-values, and the vertical axis is labeled PX. The values on the horizontal axis range from 0 to 3, in increments of 0.5. The values on the vertical axis range from 0 to 1, in increments of  0.1. There is one labeled statement inside the graph, that reads, X uniform on  (0,1). There is one smooth curve in the graph, labeled sum, and one set of many small circles, labeled Gaussian. They follow the same path, which begins at the bottom-left at the point (0, 0). The graph begins increasing at an increasing rate until approximately the point (1.5, 0.5), where it begins increasing at a decreasing rate until it has become a flat line at the top-right of the graph, at approximately point (3, 1). Figure four is a distribution graph. It is titled, distribution for the sum of three iid uniform random variables. The horizontal axis is labeled, x-values, and the vertical axis is labeled PX. The values on the horizontal axis range from 0 to 3, in increments of 0.5. The values on the vertical axis range from 0 to 1, in increments of  0.1. There is one labeled statement inside the graph, that reads, X uniform on  (0,1). There is one smooth curve in the graph, labeled sum, and one set of many small circles, labeled Gaussian. They follow the same path, which begins at the bottom-left at the point (0, 0). The graph begins increasing at an increasing rate until approximately the point (1.5, 0.5), where it begins increasing at a decreasing rate until it has become a flat line at the top-right of the graph, at approximately point (3, 1).
Distribution for the sum of three iid uniform random variables.
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For the sum of only three random variables, the fit is remarkably good. This is not entirely surprising, since the sum of two gives a symmetric triangulardistribution on ( 0 , 2 ) . Other distributions may take many more terms to get a good fit. Consider the following example.

Sum of eight iid random variables

Suppose the density is one on the intervals ( - 1 , - 0 . 5 ) and ( 0 . 5 , 1 ) . Although the density is symmetric, it has two separate regions of probability. From symmetry, E [ X ] = 0 . Calculations show Var [ X ] = E [ X 2 ] = 7 / 12 . The MATLAB computations are:

tappr Enter matrix [a b]of x-range endpoints [-1 1] Enter number of x approximation points 200Enter density as a function of t (t<=-0.5)|(t>=0.5) Use row matrices X and PX as in the simple case[z,pz] = diidsum(X,PX,8);VX = 7/12; F = cumsum(pz);FG = gaussian(0,8*VX,z); plot(z,F,z,FG)% Plottting details (see [link] )
Figure five is a distribution graph. It is titled, distribution for sum of eight iid random variables. The horizontal axis is labeled, x-values, and the vertical axis is unlabeled. The values on the horizontal axis range from -8 to 8 in increments of 2, and the values on the vertical axis range from 0 to 1 in increments of 0.1. The figure contains a second title inside the graph, which reads, Density  = 1 on (-1, -0.5) and (0.5, 1). There are two plots in this figure. The first is a solid line, labeled sum. the second is a dashed, smooth line, labeled gaussian. Both follow the same general shape, except that the solid line is not as smooth, with multiple places along its plot where it is wiggly, as if it is attempting to follow the same path as the gaussian plot but does so only with some imperfection. The gaussian pot is smooth and consistent. The shape of both plots can be described as the following. The plots begin at the bottom-left corner of the graph, at point (-8, 0) and continue to the right horizontally with negligible slope, until point (-6, 0), where the plot begins increasing at an increasing rate. It does so until the midpoint in the graph, approximately (0, 0.5), where it begins to increase at a decreasing rate as it approaches the top-right corner of the graph. By approximately (6, 1) the plot continues horizontally to the top-right corner, (8, 1). Figure five is a distribution graph. It is titled, distribution for sum of eight iid random variables. The horizontal axis is labeled, x-values, and the vertical axis is unlabeled. The values on the horizontal axis range from -8 to 8 in increments of 2, and the values on the vertical axis range from 0 to 1 in increments of 0.1. The figure contains a second title inside the graph, which reads, Density  = 1 on (-1, -0.5) and (0.5, 1). There are two plots in this figure. The first is a solid line, labeled sum. the second is a dashed, smooth line, labeled gaussian. Both follow the same general shape, except that the solid line is not as smooth, with multiple places along its plot where it is wiggly, as if it is attempting to follow the same path as the gaussian plot but does so only with some imperfection. The gaussian pot is smooth and consistent. The shape of both plots can be described as the following. The plots begin at the bottom-left corner of the graph, at point (-8, 0) and continue to the right horizontally with negligible slope, until point (-6, 0), where the plot begins increasing at an increasing rate. It does so until the midpoint in the graph, approximately (0, 0.5), where it begins to increase at a decreasing rate as it approaches the top-right corner of the graph. By approximately (6, 1) the plot continues horizontally to the top-right corner, (8, 1).
Distribution for the sum of eight iid uniform random variables.
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Although the sum of eight random variables is used, the fit to the gaussian is not as good as that for the sum of three in Example 4 . In either case, the convergence is remarkable fast—only a few terms are needed for good approximation.

Convergence phenomena in probability theory

The central limit theorem exhibits one of several kinds of convergence important in probability theory, namely convergence in distribution (sometimes called weak convergence). The increasing concentration of values of the sample average random variable A n with increasing n illustrates convergence in probability . The convergence of the sample average is a form of the so-called weak law of large numbers . For large enough n the probability that A n lies within a given distance of the population mean can be made as near one as desired. The fact that the variance of A n becomes small for large n illustrates convergence in the mean (of order 2).

E [ | A n - μ | 2 ] 0 as n

In the calculus, we deal with sequences of numbers. If { a n : 1 n } is a sequence of real numbers, we say the sequence converges iff for N sufficiently large a n approximates arbitrarily closely some number L for all n N . This unique number L is called the limit of the sequence. Convergent sequences are characterized by the fact that for largeenough N , the distance | a n - a m | between any two terms is arbitrarily small for all n , m N . Such a sequence is said to be fundamental (or Cauchy ). To be precise, if we let ϵ > 0 be the error of approximation, then the sequence is

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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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