Probability, Statistics & Random Processes Chapter 9 Solution Manual – 2025 Regression & Correlation Analysis Solutions

CHAPTER 9
g

, PROBLEMg9.1

Determinegbygdirectgintegrationgthegmomentgofginertiagofgthegshadedgarea
gwithgrespectgtogthegygaxis.




SOLUTION




Atg xg=g0,g g yg=gbg:g g bg=gkg(0g−g
)2 bg
kg =g yg=g 2 (gxg−ga) ;gdAg=g y
2g
2
then
b a a

dIgyg =g x2dAg=g x2gyg
dx agg 2g

Igyg =g 0gb a2 x xg−ga  dx
2

g 
 ag g b
=g 0gg a x −g2ax +ga2g x


4 3 2
dx
g 
2
bg g 1 a2gx3g
a
= 21 x15g −
a g g ax g 3 
4
0
 +
5
1g
Igy g =g a3bg
30




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, PROBLEMg9.2

Determineg byg directg integrationg theg momentgofg inertiagofg theg shadedg ar
eagwithgrespectgtogthegygaxis.




SOLUTION

yg=gkx1/3

Forg xg=ga:



bg 1/3
Thus: yg= x
a1/3

dIgyg =g x2dAg=g x2gyd
x
b bg
dIy =gx2 1/3 x1/3dxg= 1/3 x7/3dx
a a
b a b g 3  3g
=
 a 
Iy =g g dIyg g 1/3
0
x7/3dxg = 1/3  a10/3 
a  10 
Igy g =g a3bg
10




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HillgEducation.gPermissiongrequiredgforgreproductiongorgdisplay.

, PROBLEMg9.3

Determineg byg directg integrationg theg momentgofg inertiagofg theg shadedg ar
eagwithgrespectgtogthegygaxis.




SOLUTION

hg
Bygobservation yg=g x
gb

gNow dIgyg =gx2 dAg=g x2[(hg−g y)dx]
 xg
=ghx2 1g−  dx
 bg
b  xg
Then  
Igy g =g g dIyg g = hx 2 1g− gdx
0  bg
b
g1 3 x4g  1
=ghgg g x g  or Igyg = b3hg◀
− g
3 4bg0 12




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