Math331, Fall 2008
Instructor: David Anderson
Section 6.3 Homework Answers
Homework: pgs. 254 - 255, #โs 1, 2, 4.
1. Differentiating shows that the density of Xis given by
f(x) = ๎32xโ3xโฅ4
0x < 4.
Therefore, the expected value is
E[X] = Zโ
โโ
xf(x)dx =Zโ
4
xร32xโ3dx
= 32 Zโ
4
xโ2dx = 321
4
= 8.
The variance does not exist because
E[X2] = Zโ
4
x232 โxโ2dx =Zโ
4
32dx =โ,
and so V ar(X) = E[X2]โE[X]2does not exist.
2. We are given that
f(x) = ๎6(xโ1)(2 โx) if 1 < x < 2
0 else
Therefore,
E[X] = Zโ
โโ
xf(x)dx =Z2
1
xร6(xโ1)(2 โx)dx
=โZ2
1๎6x3โ18x2+ 12x๎dx
=โ3
2x4+ 6x3โ6x2๎
๎
๎
๎
2
x=1
=3
2.
The second moment of Xis
E[X2] = Z2
1
x2ร6(xโ1)(2 โx)dx =23
10.
Therefore, V ar(X) = 23/10 โ9/4 = 1/20. Thus, the standard deviation is
ฯX=pV ar(X) = 1
โ20.
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