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# Tutorial7_soln.pdf

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University of Waterloo

Statistics

STAT 330

Christine Dupont

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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