Please answer all 3 questions and show all work. Thank you! Define S_n = 1/n sig

Question

Please answer all 3 questions and show all work. Thank you!

Define S_n = 1/n sigma^n_i = 1 X_i where X_i's are identically and independently distributed (IID) as Uniform [a, b]. Central Limit Theorem: Let X_i's be IID rv's 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 n 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^z_infinity 1/Squareroot 2 pi e^-y^2/2 dy. In this homework you will run a program to observe the Central Limit Theorem. 1. Plot the normal CDF using its closed form expression. 2. Repeat the following for "a = -1, b = -1", "a = -5, b = 5", "a = -10, b = 10". For X_i ~ Uniform [a, b], run a program to simulate rv S_n - n mu/Squareroot n sigma and plot its simulated CDF for large enough n until it starts to resemble the normal CDF. Such as the following plot Comment on why you need larger n for the CDF to converge as b -a increases.

Explanation / Answer

When sample size is small then the standard deviation is larger, because we obtained standard deviation dividing by square root of sample size.

we can see this in above figure when n = 10

So when n increases, the CDF of Zn slowly starts to resemble the normal CDF.

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