STAT 241/541 Homework 12 Solution
Prepared by James Hu
Problem 1: Ch 10.3:10
a) Since the density function is an even function over the range (−∞,+∞),
the mean is 0. This can also be seen from the following, i.e., by symmetry,the
integrand is an odd function and the range of the integral is also symmetric
around 0, therefore,
E(X) = Z+∞
−∞
x·1
2e−|x|dx = 0 .
We can calculate the variance by using the fact V ar(X) = E(X2)−[E(X)]2,
and
E(X2) = Z+∞
−∞
x21
2e−|x|dx
= 2 Z∞
0
x21
2e−xdx =Z+∞
0
x3−1e−xdx
= Γ(3) = 2! = 2
Therefore, V ar(X) = 2 −0 = 2.
b) The moment generating function for X1is
gX1(t) = E(eX1t) = Z+∞
−∞
ex1t1
2e−|x1|dx1
=1
2hZ0
−∞
e(t+1)x1dx1+Z+∞
0
e(t−1)x1dx1i
=1
2h1
t+ 1 +1
t−1i,where |t|<1
=1
1−t2,|t|<1
Similarly, we can get the other moment generating functions, for |t|<1
gSn(t) = ³1
1−t2´n
, gAn(t) = ³1
1−(t
n)2´n
, gS∗
n(t) = ³1
1−t2
2n´n
.
c) gS∗
n(t)→et2
2as n→ ∞ .
d) gAn(t)→1 as n→ ∞ .
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