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THE UNIVERSITY OF IOWA
22S:175 HW Set 2
Due at the
beginning
of class on Monday, January 31, 2011
Write legibly.
Show your steps.
Explain your reasoning.
Simplify your answers.
1.
(An extension of #4 in HW Set 1)
Let X be a random variable for which the moment
generating function, M
X
(t) = E[e
tX
], exists.
Let
and
2
denote its mean and variance,
respectively.
For j = 3, 4, 5, .
.. , let
j
=
E[(X – μ)
j
] denote its jth central moment.
Expand ln[M
X
(t)] as a
Maclaurin series
of t,
l
n
M
X
(t)
=
0
j
j
t
j
/ j! .
Determine the first five coefficients,
0
,
1
, … ,
4
, in terms of μ,
2
, μ
3
and μ
4
.
Remarks
:
(i)
is the Greek letter
kappa
.
(ii) The function ln M
X
(t) is called K(t) in
Exercise 79 on page 95 of Ross.
(iii) The following result is very useful in actuarial
science: A random variable X is
normal
if and only if
j
= 0 for all j
≥
3.
(iv) A Maclaurin series is a special case of the Taylor series.
2.
Let X and Y be two independent
exponential
random variables with mean
X
and
Y
,
respectively.
Evaluate the conditional expectation E[X  X < Y].
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 Spring '08
 Tang,Q

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