Define S_n = 1/n sigma_i=1^n X_t \'s are identically and Independently distribut

Question

Define S_n = 1/n sigma_i=1^n X_t 's are identically and Independently distributed (IID) as Uniform[a, b] Central Limit Theorem: Let X_t's be IID rvs with finite mean mu and finite variance sigma^2.Then for every real number z. lim n rightarrow infinity Pr{S_n n mu/Squareroot pi sigma lessthanorequalto z}= Phi(z) where Phi (z) is the normal CDF, i.e. Gaussian distribution with mean 0 and variance 1 Phi(z) = integral_infinity^z 1/Squareroot 2 pi. E^-y^2/x dy In this homework you will run a program to observe the Central Limit Theorem. Plot the normal CDF using its closed form expression. Repeat the following for "a = -1.b=1" a = -5, b= 5'', 'a =10 b = 10 For X_i tilde Uniform[a, b], run a program to simulate rv and plot its Simulated CDF for large enough n until it starts to the normal CDF. Such as the following plot in "Stochastic Processes: Theory for Applications" by Robert G. Gallager Comment on why you need larger n for the CDF to converge as b-a increases.

Explanation / Answer

Normal distribution is symmetric and mesokurtic . So, the probability, ie. the area under the curve is bell shaped and it's compact in probability. For large value of n, the distribution want to be its compact format. Besides, range of Z_n will -infinite to + infinite when n tends to infinite, which is the range of normal distribution. For the uniform(a,b) distribution , if b-a increases then it wants to the form of normal distribution. That's it....

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