STAT330 Lecture Notes - Lecture 4: Random Variable, Improper Integral

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11 Aug 2016
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9.We will prove this result assuming Xis a continuous r.v. The proof for Xa discrete r.v. follows in a similar
manner with intergrals replaced by sums.
Suppose Xhas p.d.f. f(x) and E|X|kexists for some kZ+. Then the improper integral
Z
−∞
|x|kf(x)dx
converges. Let A={x:|x| ≥ 1}. Then
Z
−∞
|x|kf(x)dx =Z
A
|x|kf(x)dx +Z
¯
A
|x|kf(x)dx.
Since
0≤ |x|kf(x)f(x) for x¯
A
we have
0Z
¯
A
|x|kf(x)dx Z
¯
A
f(x)dx =PX¯
A1.(1)
Convergence of R
−∞ |x|kf(x)dx and (1) imply the convergence of R
A
|x|kf(x)dx.
Now
Z
−∞
|x|jf(x)dx =Z
A
|x|jf(x)dx +Z
¯
A
|x|jf(x)dx, j = 1,2, ..., k 1 (2)
and
0Z
¯
A
|x|jf(x)dx 1
by the same argument as in (1). Since R
A
|x|kf(x)dx converges and
|x|kf(x)≥ |x|jf(x) for all xAand j= 1,2, ..., k 1
then by the Comparison Theorem for Improper Integrals R
A
|x|jf(x)dx converges. Since both integrals on the right
side of (2) exist, therefore
E|X|j=
Z
−∞
|x|jf(x)dx exists for j= 1,2, ..., k 1.
33.Since Xvχ2(n) and U=X+Yvχ2(m) the m.g.f. of Xis
MX(t) = 1
(1 2t)n/2, t < 1
2
and the m.g.f. of Uis
MU(t) = 1
(1 2t)m/2, t < 1
2.
The m.g.f. of Ucan also be obtained as
MU(t) = EetU =Eet(X+Y)
=EetX EetY since Xand Yare independent r.v.’s
=MX(t)MY(t).
1
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