Question

Ex:- Generation of Variables by the centrad limit theorem. of independent Gaussion random aseguence 1. Suppose , we have agenerator of uni formly distributed in (o,1) random Jaria bles (such. Variates are often simply Called random numbers in (0,1)). Deduce fram approximation of the Centeral limit thearem, Un of independent a agenerator of random variables with ncan sequence Xn of independent Gaussian oistributed and variance o? from the e Un Comment on this approach 12 Preyuently used. Explian why Seguence Values of Un are Successive e seguence Un is rephaced by seguence V of independ ent unifor mly distributed in f0, 13 random. frequently 2. Sameguesdion. where the where 30 successive voalues of Vn are Variables used.

Answer 1

This Demonstration illustrates the central limit theorem for the continuous uniform distribution on an interval. If has the uniform distribution on the interval and is the mean of an independent random sample of size from this distribution, then the central limit theorem says that the corresponding standardized distribution approaches the standard normal distribution as . Using operations on the characteristic function of we can compute the PDF of more easily than we could directly. The blue curve is the PDF of and the dashed curve is the PDF of a standard normal distribution.