# 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 k∈Z+. 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

0≤Z

¯

A

|x|kf(x)dx ≤Z

¯

A

f(x)dx =PX∈¯

A≤1.(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

0≤Z

¯

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 x∈Aand 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).

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