By K. P. N. Murthy

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**Example text**

Not all distributions have this property. For example, when you add two uniform random variables, you get a random variable with triangular distribution. When two exponential random variables are added we get one with a Gamma distribution, as we would see in the next section where we shall be investigating the behaviour of the sum of several independently distributed exponential random variables. (In fact we went a step further and found that when we add N identically distributed independent random variables with finite variance, the resulting distribution tends to a Gaussian, when N → ∞.

Accordingly, a particular realization of the random variable Y¯N will lie outside the interval (−ǫ, +ǫ) with a probability less than or equal to σ 2 /(N ǫ2 ). Thus, as ǫ becomes smaller, by choosing N adequately large we find that a realization of Y¯N can be made to be as close to the mean as we desire with a probability very close to unity. This leads us naturally to the laws of large numbers. Of the several laws of large numbers, discovered over a period two hundred years, we shall see perhaps the earliest version, see Papoulis [9], which is, in a sense, already contained in the Chebyshev inequality.

A) the circular density and the C× the bounding function. (b) random points that uniformly fill up the rectangular box. (c) points that get accepted in the sampling procedure. (d) the distribution of the accepted values of x. Let me illustrate the rejection technique by considering the circular probability density function 4 f (x) = 1 − x2 for 0 ≤ x ≤ 1. (105) π A convenient choice of the bounding function is g(x) = 1 ∀ 0 ≤ x ≤ 1. Thus sampling from g(x) is equivalent to setting xi = ξi . The value of C is 4/π.

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