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STAT 330 (53)
Lecture

# Tutorial7_soln.pdf

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School
University of Waterloo
Department
Statistics
Course
STAT 330
Professor
Christine Dupont
Semester
Fall

Description
Stat 330 - Tutorial 7 Solution 1. ▯(e ▯1) Y j▯ = ▯ ▯ Poi(▯) =) E(Y j▯ = ▯) = ▯; V ar(Y j▯ = ▯) = ▯; M Y j▯=▯t) = e t =) E(Y j▯) = ▯; V ar(Y j▯) = ▯; M Y j▯t) = e(e ▯1) ▯ ▯ Gamma(▯;▯) =) E(▯) = ▯▯; V ar(▯) = ▯▯ ; M (t) = ▯1 ▯ ▯t) ▯▯ Therefore, E(Y ) = E[E(Y j▯)] = E[▯] = ▯▯ V ar(Y ) = E[V ar(Y j▯)] + V ar[E(Y j▯)] = E[▯] + V ar[▯] = ▯▯ + ▯▯ 2 = ▯▯(1 + ▯) 2. A negative binomial distribution with parameters k (k = 0;1;:::) and p (0 < p < 1) has m.g.f. ▯ ▯▯ p 1 ▯ (1 ▯ p)e t M (t) = E[etY] = E[E(etYj▯)] = E[M (t)] = E[ee ▯1] = M (e ▯ 1) = [1 ▯ ▯(e ▯ 1)] ▯▯ Y ▯ ▯ Y j▯ ▯ 1 ▯ = (1 + ▯) ▯ ▯e ▯ ▯▯ 1=(1 + ▯) = [(1 + ▯) ▯ ▯e ]=(1 + ▯) ! 1 ▯ = 1+▯ 1 ▯ ▯ et 1+▯ which is the m.g.f of a negative binomial distribution with parameters k = ▯ and p = 1 1+▯. ▯ 3. Let Mn= max(X ;1::;X )n Then the c.d.f. ofnM ns ▯▯ ▯ h ▯nx ▯in Fn(x) = P M n x = F X n ▯ where FX is the c.d.f. of the Cauchy distribution. The limiting distribution: h ▯ ▯i nx n 1 lim Fn(x) = lim FX (Indeterminate form: 1 ) n!1 n!1 n ▯ ▯ ▯o nx = exp n!1m nlogFX ▯ 1 Consider the exponent: ▯ ▯ nx n!1m nlogF (Indeterminate form: 1 ▯ 0) ▯▯ ▯ log(FX ▯x ) 0 = lim (Indeterminate form: ) n!1 1=n 0 d ▯nx▯ dn log(X ▯ ) = n!1 d (L’Hopital’s rule) dn1=n
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