Process of repeated convolution, Engineering Mathematics

If x and y are two independent random variables then their joint density function is given by

699_Process of repeated convolution.png

The density function fz of the sumĀ  of these two variables is given by the convolution

986_Process of repeated convolution1.png

The proof of this relation may be found in any good introductory text on probability. According to the preceding exercise, if x and y are each U[0,1] then their joint density is triangular, i.e., t = rlr, using symbol t for triangle. What has the central limit theorem got to do with this process of repeated convolution?

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