Generalizing Exercise 5-c for r.v.’s X_i having a mean of m and a variance of σ², find the limit

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Exercise 5-c

Let X₁,X₂, ... be a sequence of independent r.v.’s uniformly distributed on the interval [0,2], and let S_n = X₁ + ... + X_n.

(a) Find E{S_n} and Var{S_n}.

(b) Using the Central Limit Theorem, estimate P(n − √n ≤ S_n ≤ n + √n).

(c) Write down a formula for the limit of

Proceeding from the formula you have obtained, explain why the accuracy with which the average X̅_n = S_n is close to 1 has an order of 1/√n.

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P m AI N