Consider a sequence of independent standard exponential r.v.’s X₁,X₂, ..., and the r.v. S_n = X₁+...+X_n. Let Γ_aν(x) be the d.f. of a Γ-r.v. with parameters (a, ν), and as usual, f_aν(x) be the corresponding density.

(a) Show that P(S_n ≤ x) = Γ_1n(x), and the d.f. of the r.v. S^∗_n is F^∗_n(x) = Γ_1n(x√n + n).

(b) Show that the density of the r.v.

(c) Using software, for example Excel, provide a worksheet including columns with the values of the standard normal density ϕ(x) and d.f. Φ(x), and the density f ^∗_n(x) and the d.f. F^∗_n (x). It suffices to consider −3 ≤ x ≤ 3, and n equal to, say, 10,20,100. Consider the accuracy of the normal approximation; in particular, max_x |F^∗_n (x)−Φ(x)|.

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S is f (x)=nfin(xn+n).