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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 24 "Why the Normal Function?
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 417 "The \+
purpose of this laboratory is to look at some probability density func
tions for sums of many independent random numbers.  In theory this is \+
easy but in practice the kinds of integrals involved are difficult for
 calculator- and computer-based integration routines to work with.  Fo
r this reason we define and use some special purpose procedures of our
 own.  Your browser window explains the procedures that we use." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "As an exa
mple we work with the PDF" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
257 "" 0 "" {TEXT -1 31 "Psi(x) = 1/2,   if -1 <= x <= 1" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 26 "Evaluate the next
 cell now" }{TEXT -1 25 ".  It defines this pdf.  " }{TEXT 257 67 "Thi
s is the only cell that you will need to modify for this module." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "psi := x -> 1/2:\na := -1:\nb :=  1:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The next step is to repre
sent this initial pdf as a table of values.  " }{TEXT 258 27 "Evaluate
 the next cell now." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 330 "n := 50:                   # Use 50 samp
le points in table.\nh := evalf((b - a)/n):     # Interval width for t
able.\ntotal := sum(evalf(psi(a + j * h)),j=0..n):\nftable := [seq(eva
lf(psi(a + j * h))/total,j=0..n)]:\ngtable := [seq([a + j*h, ftable[j+
1]],j=0..n)]:\n\nplot(gtable,x=a..b,y=0..max(op(ftable)),xtickmarks=0,
ytickmarks=0);\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 35 "The next cell defines a procedure  " }{TEXT 259 
8 "convolve" }{TEXT -1 54 "  that computes the convolution of a pdf wi
th itself. " }{TEXT 260 27 " Evaluate the next cell now" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "convolve := pdf -> [seq(\n   sum(pdf[j+1] * pdf[k - j
 + 1],j=max(0,k-n)..min(n, k)),k=0..2*n)]:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The next cell computes th
e pdf for the sum of two independent random numbers of this kind.  " }
{TEXT 261 16 "Evaluate it now." }{TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 182 "a := a + a:\nb := b + b:\nftable := convolve(
ftable):\nn := n + n:\n\ngtable := [seq([a + j*h, ftable[j+1]],j=0..n)
]:\n\nplot(gtable,x=a..b,y=0..max(op(ftable)),xtickmarks=0,ytickmarks=
0);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 154 "The next cell repeats the previous cell and computes and
 graphs the pdf for the sum of four independent random numbers like ou
r original random numbers.  " }{TEXT 262 16 "Evaluate it now." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 182 "a := a + a:\nb := b + b:\nftable := convolve(ftable):\nn := n +
 n:\n\ngtable := [seq([a + j*h, ftable[j+1]],j=0..n)]:\n\nplot(gtable,
x=a..b,y=0..max(op(ftable)),xtickmarks=0,ytickmarks=0);\n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "The next
 cell repeats the previous cell and computes and graphs the pdf for th
e sum of eight independent random numbers like our original random num
bers.  " }{TEXT 263 16 "Evaluate it now." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "a := a + a:\nb := b
 + b:\nftable := convolve(ftable):\nn := n + n:\n\ngtable := [seq([a +
 j*h, ftable[j+1]],j=0..n)]:\n\nplot(gtable,x=a..b,y=0..max(op(ftable)
),xtickmarks=0,ytickmarks=0);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{MARK "0 0 0" 24 }{VIEWOPTS 1 1 0 1 1 1803 }