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Let Xi, Xz, Xyr - sequence of independent identically distributed random variables with zero mean and finite variance. Show that the characteristic function of Sv Yno" XN Xk can be written as: @sn (@) (ai" 4" Ax). The Taylor formula f(x) = f(0) + is valid for very small x. When N in (a) is very large, apply Taylor formula for @x (WNaz) and use the relation between the characteristic function and moments to show that: @sx (@). Use Jin (1+4" e) to find Jie_ WS, (6). Parts (a)-(e) prove the Central Limit Theorem. Parts (d) and (e) use software, but submit. (d) Let Xy, Xz, Xyr be a sequence of independent identically distributed Uniform random variables on the interval [-1,1]. Write code to verify the Central Limit Theorem by plotting the histogram for S5, S10, Sz0. Use enough sample points (i.e. realization of the sequence X; Xz, X3, ..., Xv) to generate clear plots. (e) Let X, Xz, Xy be a sequence of independent identically distributed Cauchy random variables on the interval [0,âˆž). Write code to plot the histogram for S,, S10, Sz0. Does the Central Limit Theorem hold in this case? Hint: There is built-in code for the Cauchy distribution in MATLAB. Instead, you can use the following code for part (e). Also, the same code can be modified slightly and used for part (d): Clear N = 5:810, for I = 1: 1000 (n) = 4Lm TtLam (pi (rand(1,N) 51)) {sgrL (N) end histogram(C) axis([-1000 1000 500])

          Let Xi, Xz, Xyr - sequence of independent identically distributed random variables with zero mean and finite variance.
Show that the characteristic function of Sv Yno" XN Xk can be written as: @sn (@) (ai" 4" Ax). The Taylor formula f(x) = f(0) + is valid for very small x. When N in (a) is very large, apply Taylor formula for @x (WNaz) and use the relation between the characteristic function and moments to show that: @sx (@). Use Jin (1+4" e*) to find Jie_ WS, (6).
Parts (a)-(e) prove the Central Limit Theorem.
Parts (d) and (e) use software, but submit.
(d) Let Xy, Xz, Xyr be a sequence of independent identically distributed Uniform random variables on the interval [-1,1]. Write code to verify the Central Limit Theorem by plotting the histogram for S5, S10, Sz0. Use enough sample points (i.e. realization of the sequence X; Xz, X3, ..., Xv) to generate clear plots.
(e) Let X, Xz, Xy be a sequence of independent identically distributed Cauchy random variables on the interval [0,âˆž). Write code to plot the histogram for S,, S10, Sz0. Does the Central Limit Theorem hold in this case?
Hint: There is built-in code for the Cauchy distribution in MATLAB. Instead, you can use the following code for part (e). Also, the same code can be modified slightly and used for part (d):
Clear N = 5:810,
for I = 1: 1000
(n) = 4Lm
TtLam (pi * (rand(1,N)
51)) {sgrL (N)
end
histogram(C) axis([-1000 1000 500])

letxi xz xyr sequence uf independent identically distributed random variables with zero mea and finite variance show that the characteristic functiun of sv yno xn xk be written a5 sn ai 4 ax 63936

Added by Alfonso F.

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