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{SECT 0 {EXCHG {PARA 256 "" 0 "top" {TEXT -1 28 "Sums of Two, Three, a
nd Four" }{TEXT 262 1 " " }{TEXT 289 12 "Independent " }{TEXT 288 1 "U
" }{TEXT -1 24 "-shaped Random Variables" }}}{EXCHG {PARA 3 "" 0 "" 
{TEXT -1 5 "Let  " }{XPPEDIT 18 0 "X[1],X[2];" "6$&%\"XG6#\"\"\"&F$6#
\"\"#" }{TEXT -1 37 " be a random sample of size 2 from a " }{TEXT 
263 1 "U" }{TEXT -1 35 "-shaped distribution with p.d.f. f(" }{TEXT 
265 1 "x" }{TEXT -1 9 ") = (3/2)" }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"
#" }{TEXT 264 0 "" }{TEXT -1 7 ", -1 < " }{TEXT 266 1 "x" }{TEXT -1 9 
" < 1. Let" }}{PARA 3 "" 0 "" {XPPEDIT 18 0 "Y[1];" "6#&%\"YG6#\"\"\"
" }{TEXT -1 4 " =  " }{XPPEDIT 18 0 "X[1]+X[2];" "6#,&&%\"XG6#\"\"\"F'
&F%6#\"\"#F'" }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {XPPEDIT 18 0 "Y[2];" 
"6#&%\"YG6#\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "X[2];" "6#&%\"XG6#
\"\"#" }{TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT -1 26 "Find the joint p
.d.f. of  " }{XPPEDIT 18 0 "Y[1];" "6#&%\"YG6#\"\"\"" }{TEXT -1 5 " an
d " }{XPPEDIT 18 0 "Y[2];" "6#&%\"YG6#\"\"#" }{TEXT -1 22 ".  Then int
egrate out " }{XPPEDIT 18 0 "y[2];" "6#&%\"yG6#\"\"#" }{TEXT -1 23 " t
o find the p.d.f. of " }{XPPEDIT 18 0 "Y[1];" "6#&%\"YG6#\"\"\"" }
{TEXT -1 13 ", the sum of " }{XPPEDIT 18 0 "X[1];" "6#&%\"XG6#\"\"\"" 
}{TEXT -1 5 " and " }{XPPEDIT 18 0 "X[2];" "6#&%\"XG6#\"\"#" }{TEXT 
-1 1 "." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 132 "A procedure for findi
ng the p.d.f. of the sum of two continuous random variables is given. \+
And then this is illustrated graphically." }}}{EXCHG {PARA 4 "" 0 "" 
{TEXT -1 18 "Let the p.d.f. of " }{TEXT 256 3 " X " }{TEXT -1 30 " be \+
positive on the interval [" }{TEXT 257 4 "a, b" }{TEXT -1 9 "], where \+
" }{TEXT 258 1 "a" }{TEXT -1 5 " and " }{TEXT 259 1 "b" }{TEXT -1 172 
" are finite. The following procedure will find the p.d.f. of the sum \+
of a random sample of size 2 from this distribution in which the p.d.f
. is defined as an expression in " }{TEXT 261 2 "x." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 0 "" }{TEXT 271 58 "A convolution f
ormula. [See Exercise 5.2-12(c), page 233.]" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 253 "SumX1X2 := proc(expr::algebraic, a, b);\npiecewise(y
[1] < 2*a, 0, y[1] < a + b, int(subs(x = y[1] - y[2], expr)*subs(x = y
[2], expr), y[2] = a .. y[1] - a), y[1] < 2*b, int(subs(x = y[1] - y[2
], expr)*subs(x = y[2], expr), y[2] = y[1] - b .. b), 0);\nend;" }}}
{EXCHG {PARA 4 "" 0 "" {TEXT -1 2 "A " }{TEXT 260 1 "U" }{TEXT -1 14 "
-shaped p.d.f." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "f := 3/2
*x^2;\nk := 'k':\nxtics := [-1+0.2*k$k=0..10]:\nytics := [0.5*k$k=1..3
]:\nplot(f, x = -1 .. 1, xtickmarks=xtics, ytickmarks=ytics, color=blu
e, thickness=2, labels=[``,``],font=[COURIER,BOLD,12]);" }{TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 267 68 "Using the convolution formula, we find the p.d.f. \+
of the sum of two " }{TEXT 268 1 "U" }{TEXT 269 25 "-shaped random var
iables." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 348 "pdf := SumX1X2(
f, -1, 1):\npdf := simplify(SumX1X2(f, -1, 1));\nxtics := [-2 + 0.4*k$
k=0..10]:\nytics := [0.2*k$k=1..4]:\nplot(pdf, y[1] = -2 .. 2, y=0 .. \+
0.92, xtickmarks=xtics, ytickmarks=ytics, color=blue, thickness=2,labe
ls=[``,``], title = `p.d.f. for the Sum of Two U-shaped Random Variabl
es`,font=[COURIER,BOLD,12], titlefont=[COURIER,BOLD,12]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 0 "" }{TEXT 273 72 "The follo
wing illustrates graphically what the convolution formula does." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "g := f*subs(x = y, f);   # \+
product of two p.d.f.'s\nplot3d(g, x = -1 .. 1, y = -1 .. 1, axes=boxe
d, orientation=[-66,60],title = `Joint p.d.f. of Two U-shaped Random V
ariables`, font=[COURIER,BOLD,12], titlefont=[COURIER,BOLD,12]);" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 278 "h := subs(x = x - y, f)*subs(x = y, f);  # Integrating out y is
 the convolution formula\nplot3d(h, x = y - 1 .. y + 1, y = -1 .. 1, a
xes=boxed, orientation=[-90, 0], title = `Domain of Integration for th
e Convolution Formula`,font=[COURIER,BOLD,12], titlefont=[COURIER,BOLD
,12]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 290 109 "The ab
ove graph gives only the domain of integration. The following graph sh
ows the surface over this domain." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 675 "h := subs(x = x - y, f)*subs(x = y, f);  # Plot h ov
er the domain of integration\nhplot := plot3d(h, x = y - 1 .. y + 1, y
 = -1 .. 1, axes=boxed, orientation=[-66,60], title = `Integrate over \+
y to find the p.d.f. of the\\nSum of 2 U-shaped Random Variables`, tit
lefont=[COURIER,BOLD,12]):\nt := 't':\nS1 := spacecurve([0,-1,t],t=0..
9/4,axes=boxed, orientation=[-66,60],color=black):\nS2 := spacecurve([
0,t,9/4],t=-1..1,axes=boxed, orientation=[-66,60],color=black):\nS3 :=
 spacecurve([0,1,t],t=0..9/4,axes=boxed, orientation=[-66,60],color=bl
ack):\nS4 := spacecurve([0,t,0],t=-1..1,axes=boxed, orientation=[-66,6
0],color=black):\ndisplay(\{hplot,S1,S2,S3,S4\}, font=[COURIER,BOLD,12
]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "g := simplify(piece
wise(x < -2,0, x < 0, int(h, y = -1 .. x + 1), x < 2, int(h, y = x - 1
 .. 1),0));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "plot(g, x =
 -2 .. 2, color=blue, thickness=2, xtickmarks=xtics, ytickmarks=ytics,
 labels=[``,``], title = `Graph of the p.d.f. of the Sum of Two U-shap
ed Random Variables`, font=[COURIER,BOLD,12], titlefont=[COURIER,BOLD,
12]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 274 0 "" }{TEXT 
275 39 "Using g, the p.d.f. for the sum of two " }{TEXT 276 1 "U" }
{TEXT 277 55 "-shaped random variables and f, the p.d.f. of a single \+
" }{TEXT 278 1 "U" }{TEXT 279 46 "-shaped random variable, plot the jo
int p.d.f." }{TEXT -1 1 " " }{TEXT 280 0 "" }{TEXT 281 18 "of the sum \+
of two " }}{PARA 0 "" 0 "" {TEXT 282 1 "U" }{TEXT 283 25 "-shaped rand
om variables " }{TEXT 291 1 "(" }{XPPEDIT 292 0 "X[1]+X[2];" "6#,&&%\"
XG6#\"\"\"F'&F%6#\"\"#F'" }{TEXT 293 1 ")" }{TEXT -1 1 " " }{TEXT 284 
0 "" }{TEXT 285 12 "and a third " }{TEXT 286 1 "U" }{TEXT 287 24 "-sha
ped random variable " }{TEXT 294 1 "(" }{XPPEDIT 295 0 "X[3];" "6#&%\"
XG6#\"\"$" }{TEXT 296 1 ")" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 211 "plot3d(g*subs(x=y,f), x = -2 .. 2, y = -1 .. 1, ax
es=boxed, orientation=[-68,56], title = `Joint p.d.f. of the Sum of 2 \+
and of 1\\nU-shaped Random Variables`, font=[COURIER,BOLD,12], titlefo
nt=[COURIER,BOLD,12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "
h := subs(x = x - y, g)*subs(x = y, f):\nplot3d(h, x = y - 2 .. y + 2,
 y = -1 .. 1, axes=boxed, orientation=[-68,56],title = `Integrate Out \+
y to Find the p.d.f. of the\\nSum of 3 U-shaped Random Variables`, fon
t=[COURIER,BOLD,12], titlefont=[COURIER,BOLD,12]);" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 297 34 "Here is the domain
 of integration." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "h := s
ubs(x = x - y, g)*subs(x = y, f):\nplot3d(h, x = y - 2 .. y + 2, y = -
1 .. 1, axes=boxed, orientation=[-90, 0],title = `Domain of Integratio
n`, font=[COURIER,BOLD,12], titlefont=[COURIER,BOLD,12]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "h := subs(x = x - y, g)*subs(x = y,
 f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "g := simplify(piec
ewise(x <= -3,0, x < -1, int(h, y = -1 .. x + 2), x = -1, 51/140, x < \+
1, int(h, y = -1 .. 1),x = 1, 51/140, x < 3, int(h, y = x - 2 .. 1),0)
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "xtics := [-4 + 0.5*k
$k=0..16]:\nytics := [0.1*k$k=1..5]:\nplot(g, x = -4 .. 4, xtickmarks=
xtics, ytickmarks=ytics, labels=[``,``], thickness=2, color=blue, titl
e = `Graph of the p.d.f. of the Sum of Three U-shaped Random Variables
`, font=[COURIER,BOLD,12], titlefont=[COURIER,BOLD,12]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "plot3d(subs(y[1] = x, pdf)*subs(y[
1] = y, pdf), x = -2 ..2, y = -2 .. 2, axes=boxed, orientation=[-63,52
], title = `Joint p.d.f. of Sum of 2 and Sum of 2\\nU-shaped Random Va
riables`, font=[COURIER,BOLD,12], titlefont=[COURIER,BOLD,12]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "plot3d(subs(y[1] = x - y, p
df)*subs(y[1] = y, pdf), x = y - 2 .. y +2, y = -2 .. 2, grid = [119,5
9], axes=boxed, orientation=[-68,56], title = `Integrate Out y to Find
 the p.d.f. of the\\nSum of 4 U-shaped Random Variables`, font=[COURIE
R,BOLD,12], titlefont=[COURIER,BOLD,12]);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 220 "plot3d(subs(y[1] = x - y, pdf)*subs(y[1] = y, pdf)
, x = y - 2 .. y +2, y = -2 .. 2, grid = [119,59], axes=boxed, orienta
tion=[-90, 0],title = `Domain of Integration`, font=[COURIER,BOLD,12],
 titlefont=[COURIER,BOLD,12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 152 "h := subs(y[1] = x - y, pdf)*subs(y[1] = y, pdf):\ng := simpl
ify(piecewise(x <= -4,0, x < 0, int(h, y = -2 .. x + 2),  x < 4, int(h
, y = x - 2 .. 2),0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "
xtics := [-4 + 1*k$k=0..8]:\nytics := [0.05*k$k=1..10]:\nplot(g, x = -
4.5 .. 4.5, xtickmarks=xtics, ytickmarks=ytics, labels=[``,``], thickn
ess=2, color=blue, title = `Graph of the p.d.f. of the Sum of Four U-s
haped Random Variables`, font=[COURIER,BOLD,12], titlefont=[COURIER,BO
LD,12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0
 4" 24 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }