2. For n ≥ 1, let X₁, X2,..., Xn be a random sample (that is, X₁, X2,..., Xn are inde- pendent) from a geometric distribution with success probability p= 0.8.
(a) Find the mgf Mys (t) of Y₁ = X₁ + X2 + X3 + X₁ + X5 using the geometric mgf. Then name the distribution of Y5 and give the value of its parameter(s).
(b) Find the mgf My, (t) of Yn = X₁ + X₂ + + Xn for any ≥ 1. Then name the distribution of Yn and give the value of its parameter(s).
(c) Find the mgf My, (t) of the sample mean Y₁ = Y. For the next two questions, Taylor series expansion of ear and the result
lim [1 + an¹ + o(n-1)]bn = eab
n→[infinity]
may be useful.
(d) Find the limit lim, My, (t) using the result of (c). What distribution does the limiting mgf correspond to?
(e) Let
Zn = √n (yn-5/4 /√5/4) =4/5 √5nyn - √5n..
Find Mz, (t), the mgf of Zn. Then use a theoretical argument to find the limiting mgf limn→[infinity] Mz, (t). What is the limiting distribution of Zn?
