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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "restart: read `a:sta
tvr4.m`: with(plots): randomize(): with(student):" }}}{EXCHG {PARA 18 
"" 0 "" {TEXT -1 24 "Using MAPLE V Release 5 " }}{PARA 18 "" 0 "" 
{TEXT -1 23 "To Find the p.d.f.'s of" }}{PARA 18 "" 0 "" {TEXT -1 24 "
Sums of Random Variables" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}
{PARA 258 "" 0 "" {TEXT 259 15 "Elliot A.Tanis " }}{PARA 259 "" 0 "" 
{TEXT 260 62 "Department of Mathematics\nHope College\nHolland, MI  49
422-9000" }}{PARA 260 "" 0 "" {TEXT 258 14 "tanis@hope.edu" }}{PARA 
257 "" 0 "" {TEXT -1 27 "http://math.hope.edu/~tanis" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 257 8 "ABSTRACT" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 228 "Simulati
on is an effective way for students to gain understanding of the Centr
al Limit Theorem. MAPLE can do simulations. But MAPLE brings the added
 capability of being able to illustrate the Central Limit Theorem theo
retically." }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 
-1 4 "Let " }{TEXT 296 1 "Y" }{TEXT -1 18 " equal the sum of " }{TEXT 
297 1 "n" }{TEXT -1 13 " rolls of an " }{TEXT 301 1 "m" }{TEXT -1 54 "
-sided die. It is not difficult to find the p.d.f. of " }{TEXT 298 1 "
Y" }{TEXT -1 27 "  but it can be tedious if " }{TEXT 299 1 "n" }{TEXT 
-1 88 " is very large. MAPLE can do this very easily. Graphically the \+
probability histogram of " }{TEXT 300 1 "Y" }{TEXT -1 47 " can be comp
ared to a normal p.d.f. with mean  " }{XPPEDIT 18 0 "n*(m+1)/2;" "6#*(
%\"nG\"\"\",&%\"mGF%\"\"\"F%F%\"\"#!\"\"" }{TEXT -1 16 "  and variance
  " }{XPPEDIT 18 0 "n*(m^2-1)/12;" "6#*(%\"nG\"\"\",&*$%\"mG\"\"#F%\"
\"\"!\"\"F%\"#7F+" }{TEXT -1 80 " .  Animation is an effective way to \+
illustrate the convergence to normality as " }{TEXT 315 1 "n" }{TEXT 
-1 12 " increases. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "
" {TEXT -1 4 "Let " }{TEXT 303 1 "Y" }{TEXT -1 42 " equal the sum of a
 random sample of size " }{TEXT 304 1 "n" }{TEXT -1 93 " from a contin
uous uniform distribution on the interval (0, 1). MAPLE can find the p
.d.f. of " }{TEXT 305 1 "Y" }{TEXT -1 72 " and use animation to illust
rate the convergence of the distribution of " }{TEXT 306 1 "Y" }{TEXT 
-1 26 " to a normal distribution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 4 "" 0 "" {TEXT -1 8 "Now let " }{TEXT 307 1 "Y" }{TEXT -1 42 " \+
equal the sum of a random sample of size " }{TEXT 308 1 "n" }{TEXT -1 
35 " from the distribution with p.d.f. " }}{PARA 4 "" 0 "" {XPPEDIT 
18 0 "f(x) = 3*x^2/2;" "6#/-%\"fG6#%\"xG*(\"\"$\"\"\"*$F'\"\"#F*\"\"#!
\"\"" }{TEXT -1 4 " ,  " }{XPPEDIT 18 0 "-1;" "6#,$\"\"\"!\"\"" }
{TEXT -1 3 " < " }{TEXT 302 1 "x" }{TEXT -1 42 " < 1.  Again MAPLE can
 find the p.d.f. of " }{TEXT 310 1 "Y" }{TEXT -1 112 ". It is extremel
y interesting to use animation to illustrate the convergence to normal
ity because the p.d.f. of " }{TEXT 311 1 "Y" }{TEXT -1 6 " has  " }
{XPPEDIT 18 0 "n+1;" "6#,&%\"nG\"\"\"\"\"\"F%" }{TEXT -1 65 "  relativ
e maxima. To help students understand why the p.d.f. of " }{TEXT 312 
1 "Y" }{TEXT -1 90 "  has so many relative maxima, it is instructive t
o look at 3-D graphs. For example, let  " }{XPPEDIT 18 0 "U = X[1]+X[2
];" "6#/%\"UG,&&%\"XG6#\"\"\"\"\"\"&F'6#\"\"#F*" }{TEXT -1 11 "  and l
et  " }{XPPEDIT 18 0 "V = X[3]+X[4];" "6#/%\"VG,&&%\"XG6#\"\"$\"\"\"&F
'6#\"\"%F*" }{TEXT -1 28 ". Graph the joint p.d.f. of " }{TEXT 313 1 "
U" }{TEXT -1 5 " and " }{TEXT 314 1 "V" }{TEXT -1 81 ". And then graph
ically illustrate the integration needed to find the p.d.f. of   " }
{XPPEDIT 18 0 "U+V = X[1]+X[2]+X[3]+X[4];" "6#/,&%\"UG\"\"\"%\"VGF&,*&
%\"XG6#\"\"\"F&&F*6#\"\"#F&&F*6#\"\"$F&&F*6#\"\"%F&" }{TEXT -1 18 ", t
he sum of four " }{TEXT 309 1 "U" }{TEXT -1 26 "-shaped random variabl
es. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "Sums of Discrete Random \+
Variables" }}{PARA 4 "" 0 "" {TEXT -1 7 "1. Let " }{XPPEDIT 18 0 "X[1]
;" "6#&%\"XG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "X[2];" "6#&%\"X
G6#\"\"#" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "X[n];" "6#&%\"XG6#%\"n
G" }{TEXT -1 34 " denote the outcomes when rolling " }{TEXT 282 1 "n" 
}{TEXT -1 17 " fair dice.  Let " }{XPPEDIT 18 0 "Y[n] = X[1]+X[2];" "6
#/&%\"YG6#%\"nG,&&%\"XG6#\"\"\"\"\"\"&F*6#\"\"#F-" }{TEXT -1 2 " +" }
{TEXT 283 7 " ... + " }{XPPEDIT 18 0 "X[n];" "6#&%\"XG6#%\"nG" }{TEXT 
-1 25 "  equal the sum of these " }{TEXT 284 1 "n" }{TEXT -1 77 " roll
s. Use animation to plot probability histograms for the distribution o
f " }{XPPEDIT 18 0 "Y[n];" "6#&%\"YG6#%\"nG" }{TEXT -1 3 ".  " }
{HYPERLNK 17 "Solution" 1 "sumdis.mws" "top" }}{PARA 4 "" 0 "" {TEXT 
-1 8 "2.  Let " }{XPPEDIT 18 0 "X[1];" "6#&%\"XG6#\"\"\"" }{TEXT -1 2 
", " }{XPPEDIT 18 0 "X[2];" "6#&%\"XG6#\"\"#" }{TEXT -1 7 ", ..., " }
{XPPEDIT 18 0 "X[n];" "6#&%\"XG6#%\"nG" }{TEXT -1 34 " denote the outc
omes when rolling " }{TEXT 285 1 "n" }{TEXT -1 17 " fair dice.  Let " 
}{XPPEDIT 18 0 "Y[n] = X[1]+X[2];" "6#/&%\"YG6#%\"nG,&&%\"XG6#\"\"\"\"
\"\"&F*6#\"\"#F-" }{TEXT -1 2 " +" }{TEXT 256 7 " ... + " }{XPPEDIT 
18 0 "X[n];" "6#&%\"XG6#%\"nG" }{TEXT -1 25 "  equal the sum of these \+
" }{TEXT 286 1 "n" }{TEXT -1 77 " rolls. Use animation to plot probabi
lity histograms for the distribution of " }{XPPEDIT 18 0 "Y[n];" "6#&%
\"YG6#%\"nG" }{TEXT -1 31 ". Superimpose a normal p.d.f.  " }
{HYPERLNK 17 "Solution" 1 "sumdis.mws" "normal" }}{PARA 4 "" 0 "" 
{TEXT -1 7 "3. Let " }{XPPEDIT 18 0 "X[1];" "6#&%\"XG6#\"\"\"" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "X[2];" "6#&%\"XG6#\"\"#" }{TEXT -1 7 ", ...,
 " }{XPPEDIT 18 0 "X[n];" "6#&%\"XG6#%\"nG" }{TEXT -1 32 " denote a ra
ndom sample of size " }{TEXT 295 1 "n" }{TEXT -1 36 " from the distrib
ution with p.d.f.  " }}{PARA 4 "" 0 "" {TEXT 287 1 "f" }{TEXT -1 1 "(
" }{TEXT 288 1 "x" }{TEXT -1 5 ") = (" }{TEXT 289 1 "x" }{TEXT -1 9 " \+
+ 1)/6, " }{TEXT 290 2 "x " }{TEXT -1 17 "= 0, 1, 2.  Let  " }
{XPPEDIT 18 0 "Y[n] = X[1]+X[2];" "6#/&%\"YG6#%\"nG,&&%\"XG6#\"\"\"\"
\"\"&F*6#\"\"#F-" }{TEXT -1 2 " +" }{TEXT 256 7 " ... + " }{XPPEDIT 
18 0 "X[n];" "6#&%\"XG6#%\"nG" }{TEXT -1 59 ".  Use animation to show \+
that the probability histogram of " }{XPPEDIT 18 0 "Y[n];" "6#&%\"YG6#
%\"nG" }{TEXT -1 5 ", as " }{TEXT 291 1 "n" }{TEXT -1 76 " increases, \+
becomes more and more symmetric and can be approximated with a  " }
{TEXT 292 1 "N" }{TEXT -1 1 "(" }{TEXT 293 1 "n" }{XPPEDIT 18 0 "mu;" 
"6#%#muG" }{TEXT -1 2 ", " }{TEXT 294 1 "n" }{XPPEDIT 18 0 "sigma^2;" 
"6#*$%&sigmaG\"\"#" }{TEXT -1 10 ") p.d.f.  " }{HYPERLNK 17 "Solution
" 1 "sumdis.mws" "Ex5.2.15" }{TEXT -1 1 " " }}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 35 "Sums of Continuous Random Variables" }}{PARA 4 "" 0 "" 
{TEXT -1 4 "Let " }{XPPEDIT 18 0 "X[1];" "6#&%\"XG6#\"\"\"" }{TEXT -1 
2 ", " }{XPPEDIT 18 0 "X[2];" "6#&%\"XG6#\"\"#" }{TEXT -1 7 ", ..., " 
}{XPPEDIT 18 0 "X[n];" "6#&%\"XG6#%\"nG" }{TEXT -1 28 " be a random sa
mple of size " }{TEXT 261 1 "n" }{TEXT -1 45 " from a continuous distr
ibution with p.d.f.  " }{TEXT 262 4 "f(x)" }{TEXT -1 8 ", where " }
{TEXT 263 4 "f(x)" }{TEXT -1 9 " > 0 for " }{TEXT 264 1 "a" }{TEXT -1 
3 " < " }{TEXT 265 1 "x" }{TEXT -1 3 " < " }{TEXT 266 2 "b " }{TEXT 
-1 3 "and" }{TEXT 271 2 "  " }{XPPEDIT 18 0 "-infinity;" "6#,$%)infini
tyG!\"\"" }{TEXT -1 3 " < " }{TEXT 267 1 "a" }{TEXT -1 3 " < " }{TEXT 
268 2 "x " }{TEXT -1 2 "< " }{TEXT 269 1 "b" }{TEXT -1 3 " < " }
{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 105 ". This sectio
n illustrates how MAPLE can be used to find the p.d.f. of the sum of t
hese random variables." }}{PARA 4 "" 0 "" {TEXT -1 8 "1. Let  " }
{TEXT 270 4 "f(x)" }{TEXT -1 9 " = 1, 0 <" }{TEXT 272 2 " x" }{TEXT 
-1 38 " < 1. Show how to find the p.d.f. of  " }{XPPEDIT 18 0 "Y[1] = \+
X[1]+X[2];" "6#/&%\"YG6#\"\"\",&&%\"XG6#\"\"\"\"\"\"&F*6#\"\"#F-" }
{TEXT -1 3 ".  " }{HYPERLNK 17 "Solution" 1 "u_0_1.mws" "top" }}{PARA 
4 "" 0 "" {TEXT -1 8 "2. Let  " }{TEXT 273 4 "f(x)" }{TEXT -1 10 " = 1
, 0 < " }{TEXT 274 1 "x" }{TEXT -1 26 " < 1. Find the p.d.f. of  " }
{XPPEDIT 18 0 "X[1]+X[2]+X[3];" "6#,(&%\"XG6#\"\"\"\"\"\"&F%6#\"\"#F(&
F%6#\"\"$F(" }{TEXT -1 11 " + . . . + " }{XPPEDIT 18 0 "X[n];" "6#&%\"
XG6#%\"nG" }{TEXT -1 3 ".  " }{HYPERLNK 17 "Solution" 1 "fig5_2.mws" "
top" }{TEXT -1 2 "  " }}{PARA 4 "" 0 "" {TEXT -1 8 "3. Let  " }{TEXT 
275 4 "f(x)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "3*x^2/2;" "6#*(\"\"$\"
\"\"*$%\"xG\"\"#F%\"\"#!\"\"" }{TEXT -1 4 " ,  " }{XPPEDIT 18 0 "-1;" 
"6#,$\"\"\"!\"\"" }{TEXT -1 3 " < " }{TEXT 276 2 "x " }{TEXT -1 37 "< \+
1.  Show how to find the p.d.f. of " }{XPPEDIT 18 0 "X[1]+X[2];" "6#,&
&%\"XG6#\"\"\"\"\"\"&F%6#\"\"#F(" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "X[
1]+X[2]+X[3];" "6#,(&%\"XG6#\"\"\"\"\"\"&F%6#\"\"#F(&F%6#\"\"$F(" }
{TEXT -1 3 ",  " }{XPPEDIT 18 0 "X[1]+X[2]+X[3]+X[4];" "6#,*&%\"XG6#\"
\"\"\"\"\"&F%6#\"\"#F(&F%6#\"\"$F(&F%6#\"\"%F(" }{TEXT -1 3 ".  " }
{HYPERLNK 17 "Solution" 1 "ushape.mws" "top" }}{PARA 4 "" 0 "" {TEXT 
-1 8 "4. Let  " }{TEXT 277 4 "f(x)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "
3*x^2/2;" "6#*(\"\"$\"\"\"*$%\"xG\"\"#F%\"\"#!\"\"" }{TEXT -1 4 " ,  \+
" }{XPPEDIT 18 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 
18 0 "x;" "6#%\"xG" }{TEXT -1 0 "" }{TEXT 256 1 " " }{TEXT -1 26 "< 1.
  Find the p.d.f. of  " }{XPPEDIT 18 0 "X[1]+X[2]+X[3];" "6#,(&%\"XG6#
\"\"\"\"\"\"&F%6#\"\"#F(&F%6#\"\"$F(" }{TEXT -1 10 " + . . . +" }
{XPPEDIT 18 0 "X[n];" "6#&%\"XG6#%\"nG" }{TEXT -1 3 ".  " }{HYPERLNK 
17 "Solution" 1 "fig5_5.mws" "theoretical" }{TEXT -1 2 "  " }}{PARA 4 
"" 0 "" {TEXT -1 8 "5. Let  " }{TEXT 278 4 "f(x)" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "(x+1)/2;" "6#*&,&%\"xG\"\"\"\"\"\"F&F&\"\"#!\"\"" }
{TEXT -1 3 " , " }{XPPEDIT 18 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT -1 3 " \+
< " }{TEXT 279 1 "x" }{TEXT -1 39 " < 1.  Show how to find the p.d.f. \+
of  " }{XPPEDIT 18 0 "Y[1] = X[1]+X[2];" "6#/&%\"YG6#\"\"\",&&%\"XG6#
\"\"\"\"\"\"&F*6#\"\"#F-" }{TEXT -1 3 ".  " }{HYPERLNK 17 "Solution" 
1 "triang.mws" "top" }}{PARA 4 "" 0 "" {TEXT -1 8 "6. Let  " }{TEXT 
280 4 "f(x)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(x+1)/2;" "6#*&,&%\"xG
\"\"\"\"\"\"F&F&\"\"#!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "-1;" "6#
,$\"\"\"!\"\"" }{TEXT -1 3 " < " }{TEXT 281 1 "x" }{TEXT -1 26 " < 1. \+
Find the p.d.f. of  " }{XPPEDIT 18 0 "X[1]+X[2]+X[3];" "6#,(&%\"XG6#\"
\"\"\"\"\"&F%6#\"\"#F(&F%6#\"\"$F(" }{TEXT -1 10 " + . . . +" }
{XPPEDIT 18 0 "X[n];" "6#&%\"XG6#%\"nG" }{TEXT -1 3 ".  " }{HYPERLNK 
17 "Solution" 1 "fig5_4.mws" "top" }{TEXT -1 0 "" }}}}{MARK "0 0 0" 
69 }{VIEWOPTS 1 1 0 1 1 1803 }