Question

Central Limit Theorem Let $X_1, X_2,...$ be a sequence of iid random variable with finite mean $mu$ and finite variance $sigma^2$, and let $S_n$ be the sum of the first $n$ random variables in the sequence: $S_n = X_1 + X_2 + ... + X_n$. (a) Let $X_i$ be a uniform continuous random variable taking values in the interval $(0, 3)$. Write a MATLAB program to plot the pdf and cdf of $S_n$. Consider $n = 1, 2, 3, 4, 5, 10, 20, 40$ and compare your results. (b) Calculate analytically the mean and the variance of $X_i$ and of $S_n$ in part (a). (c) Write a MATLAB program to generate a Gaussian random variable with the same mean and variance as $S_n$. Superimpose this plot with the plots from part (a). (d) Repeat parts (a)-(c) with $X_i$ representing a toss of a fair 6-sided die (e) Repeat parts (a)-(c) with $X_i$ representing a toss of an unfair 6-sided die with even sides twice as likely as odd sides Use $t = 10^4$ samples in the above.

          Central Limit Theorem Let $X_1, X_2,...$ be a sequence of iid random variable
with finite mean $mu$ and finite variance $sigma^2$, and let $S_n$ be the sum of the first $n$ random
variables in the sequence:
$S_n = X_1 + X_2 + ... + X_n$.
(a) Let $X_i$ be a uniform continuous random variable taking values in the interval
$(0, 3)$. Write a MATLAB program to plot the pdf and cdf of $S_n$. Consider
$n = 1, 2, 3, 4, 5, 10, 20, 40$ and compare your results.
(b) Calculate analytically the mean and the variance of $X_i$ and of $S_n$ in part (a).
(c) Write a MATLAB program to generate a Gaussian random variable with the same
mean and variance as $S_n$. Superimpose this plot with the plots from part (a).
(d) Repeat parts (a)-(c) with $X_i$ representing a toss of a fair 6-sided die
(e) Repeat parts (a)-(c) with $X_i$ representing a toss of an unfair 6-sided die with even
sides twice as likely as odd sides
Use $t = 10^4$ samples in the above.

Central Limit Theorem Let X1, X2,... be a sequence of iid random variable
with finite mean mu and finite variance sigma^2, and let Sn be the sum of the first n random
variables in the sequence:
Sn = X1 + X2 + ... + Xn.
(a) Let Xi be a uniform continuous random variable taking values in the interval
(0, 3). Write a MATLAB program to plot the pdf and cdf of Sn. Consider
n = 1, 2, 3, 4, 5, 10, 20, 40 and compare your results.
(b) Calculate analytically the mean and the variance of Xi and of Sn in part (a).
(c) Write a MATLAB program to generate a Gaussian random variable with the same
mean and variance as Sn. Superimpose this plot with the plots from part (a).
(d) Repeat parts (a)-(c) with Xi representing a toss of a fair 6-sided die
(e) Repeat parts (a)-(c) with Xi representing a toss of an unfair 6-sided die with even
sides twice as likely as odd sides
Use t = 10^4 samples in the above.

Added by Larry A.

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