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{SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 43 "Probability Distribution \+
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{EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 256 13 "The Binomial(" }
{TEXT 336 3 "n,p" }{TEXT 337 14 ") Distribution" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The Binomial(" }{TEXT 
281 3 "n,p" }{TEXT -1 77 ") distribution is a discrete distribution on
 the set of  integers \{0,1,2,...," }{TEXT 282 1 "n" }{TEXT -1 85 "\}.
  A conceptual picture of the binomial distribution can be realized by
 considering " }{TEXT 283 1 "n" }{TEXT -1 63 " independent flips of a \+
coin that lands heads with probability " }{TEXT 330 1 "p" }{TEXT -1 6 
".  If " }{TEXT 284 1 "X" }{TEXT -1 40 " is the number of heads observ
ed in the " }{TEXT 285 1 "n" }{TEXT -1 40 " coin tosses, then the prob
ability that " }{TEXT 286 3 "X=k" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "               " }{TEXT 
287 1 "f" }{TEXT -1 2 " (" }{TEXT 288 1 "k" }{TEXT -1 6 ") = P(" }
{TEXT 289 3 "X=k" }{TEXT -1 13 ") = binomial(" }{TEXT 295 3 "n,k" }
{TEXT -1 1 ")" }{XPPEDIT 18 0 "p^k*(1-p)^(n-k)" "6#*&)%\"pG%\"kG\"\"\"
),&\"\"\"F'F%!\"\",&%\"nGF'F&F+F'" }{TEXT -1 3 " , " }{TEXT 290 2 "k \+
" }{TEXT -1 10 "= 0,1,...," }{TEXT 291 1 "n" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "where binomial(" }{TEXT 301 3 "
n,k" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "n!/(k!*(n-k)!)" "6#*&-%*factor
ialG6#%\"nG\"\"\"*&-F%6#%\"kGF(-F%6#,&F'F(F,!\"\"F(F0" }{TEXT -1 22 " \+
.  The probability P(" }{TEXT 302 3 "X=k" }{TEXT -1 37 ") is derived f
rom the facts that any " }{TEXT 331 10 "particular" }{TEXT -1 13 " seq
uence of " }{TEXT 292 1 "k" }{TEXT -1 11 " heads and " }{TEXT 293 3 "n
-k" }{TEXT -1 23 " tails has probability " }{XPPEDIT 18 0 "p^k*(1-p)^(
n-k)" "6#*&)%\"pG%\"kG\"\"\"),&\"\"\"F'F%!\"\",&%\"nGF'F&F+F'" }{TEXT 
-1 25 ", and there are binomial(" }{TEXT 294 3 "n,k" }{TEXT -1 48 ") s
uch equally likely sequences.  This variable " }{TEXT 296 1 "X" }
{TEXT -1 28 " is said to have a Binomial(" }{TEXT 297 3 "n,p" }{TEXT 
-1 20 ") distribution, and " }{TEXT 298 1 "f" }{TEXT -1 1 "(" }{TEXT 
299 1 "k" }{TEXT -1 18 ") is the Binomial(" }{TEXT 300 3 "n,p" }{TEXT 
-1 54 ") probability distribution function.  More generally, " }{TEXT 
338 1 "X" }{TEXT -1 58 " may be referred to as the total number of \"s
uccesses\" in " }{TEXT 339 1 "n" }{TEXT -1 74 " independent trials, wh
ere a success is obtaining a head on the coin flip." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The following code will draw a \+
probability histogram for the Binomial(" }{TEXT 305 3 "n,p" }{TEXT -1 
34 ") distribution for your choice of " }{TEXT 303 1 "n" }{TEXT -1 5 "
 and " }{TEXT 304 1 "p" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {MPLTEXT 0 
21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=10;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "p:=0.5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG$\"\"
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PDF(n,p,x),0..10,11);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "To
 see the effect of " }{TEXT 306 1 "n" }{TEXT -1 30 " on the shape of t
he Binomial(" }{TEXT 307 3 "n,p" }{TEXT -1 88 ") distribution, the fol
lowing animation will draw a series of probability histograms as " }
{TEXT 310 1 "n" }{TEXT -1 26 " changes from 4 to 50 and " }{TEXT 311 
2 "p=" }{TEXT -1 3 "0.4" }{TEXT 340 2 ", " }{TEXT -1 45 "but you are e
ncouraged to try other values of" }{TEXT 341 2 " p" }{TEXT -1 1 "." }}
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{MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "display([seq
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\\mFa`lFb`l-Fj`l6%F\\al%*p~is~.490GFb`lFcalFgal7&-F(6$7`uF+7$F,$F\\^oF
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4\\$F\\o7$F^oFhe\\mF`oF`o7$F^o$\"1+++/>i&p\"FZ7$FfoF\\f\\mFhoFho7$Ffo$
\"1+++b.W%*pFZ7$F^pF`f\\mF`pF`p7$F^p$\"1+++E@\"p[#FR7$FfpFdf\\mFhpFhp7
$Ffp$\"1+++\"fF%4xFR7$F^qFhf\\mFaqFaq7$F^q$\"1+++h6d-@FJ7$FgqF\\g\\mFi
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F\\h\\mFisFis7$Fgs$\"1+++X`O;dFB7$F_tF`h\\mFatFat7$F_t$\"1+++SG;q!)FB7
$FgtFdh\\mFitFit7$Fgt$\"1+++uu=J5F.7$F_uFhh\\mFauFau7$F_u$\"1+++\\l+%>
\"F.7$FguF\\i\\mFiuFiu7$Fgu$\"1+++woq`7F.7$F_vF`i\\mFavFav7$F_vF\\i\\m
7$FgvF\\i\\mFivFiv7$FgvFhh\\m7$F_wFhh\\mFawFaw7$F_wFdh\\m7$FgwFdh\\mFi
wFiw7$FgwF`h\\m7$F_xF`h\\mFaxFax7$F_xF\\h\\m7$FgxF\\h\\mFixFix7$FgxFhg
\\m7$F_yFhg\\mFayFay7$F_yFdg\\m7$FgyFdg\\mFiyFiy7$FgyF`g\\m7$F_zF`g\\m
FazFaz7$F_zF\\g\\m7$FgzF\\g\\mFizFiz7$FgzFhf\\m7$F_[lFhf\\mFa[lFa[l7$F
_[lFdf\\m7$Fg[lFdf\\mFi[lFi[l7$Fg[lF`f\\m7$F_\\lF`f\\mFa\\lFa\\l7$F_\\
lF\\f\\m7$Fg\\lF\\f\\mFi\\lFi\\l7$Fg\\lFhe\\m7$F_]lFhe\\mFa]lFa]l7$F_]
lFde\\m7$Fg]lFde\\mFi]lFi]l7$Fg]lF`e\\m7$F_^lF`e\\mFa^lFa^l7$F_^lF\\e
\\m7$Fg^lF\\e\\mFi^lFi^l7$Fg^lFhd\\m7$F__lFhd\\mFa_lFa_l7$F__lFdd\\m7$
Fg_lFdd\\mFi_lFi_l7$Fg_lFad\\m7$F_`lFad\\mFa`lFb`l-Fj`l6%F\\al%*p~is~.
500GFb`lFcalFgal-%&TITLEG6#Q-Increasing~p6\"" 1 2 0 1 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 
31 "Sample Probability Calculations" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "The cumulative distribution fu
nction (CDF) may be used to compute probabilities for binomial distrib
utions.  The CDF of the Binomial(10,0.5) is shown below." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rest
art: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "n:=10:
 p:=0.5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "PlotDiscCDF(Bin
omialPDF(n,p,x),0..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "60-%'CURVESG6#7$7$\"\"!$\"1++++]il(*!#>7$$\"\"\"F(F)-F$6#
7$7$$\"\"#F($\"1+++++voa!#<7$$\"\"$F(F5-F$6#7$7$F-$\"1+++](=U2\"F77$F3
F?-F$6#7$7$$\"\"%F($\"1+++]7`pP!#;7$$\"\"&F(FH-F$6#7$7$F9$\"1+++++v=<F
J7$FFFR-F$6#7$7$FL$\"1+++](o/B'FJ7$$\"\"'F(FY-F$6#7$7$Ffn$\"1+++++D\"G
)FJ7$$\"\"(F(F\\o-F$6#7$7$F_o$\"1++++]7`%*FJ7$$\"\")F(Feo-F$6#7$7$Fho$
\"1+++D\"yD*)*FJ7$$\"\"*F(F^p-F$6#7$7$$\"#5F(F-7$$\"#6F(F--F$6#7$7$Fap
$\"1+++vVB!***FJ7$FgpF`q-%+AXESLABELSG6$%!GFfq-%'COLOURG6&%$RGBGF(F($
\"*++++\"!\")-%%VIEWG6$%(DEFAULTGFar" 1 2 0 1 0 2 6 1 4 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "We can c
alculate probabilities for the Binomial(" }{TEXT 314 3 "n,p" }{TEXT 
-1 146 ") distribution using either the probability distribution funct
ion (PDF) or the cumulative distribution function (CDF).  We will deno
te the PDF by " }{TEXT 257 1 "f" }{TEXT -1 16 " and the CDF by " }
{TEXT 258 1 "F" }{TEXT -1 51 ".  We will consider these functions for \+
a variable " }{TEXT 315 1 "X" }{TEXT -1 42 " having the Binomial(10,0.
5) distribution." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=10;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"nG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:=0.5;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG$\"\"&!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f:=x->BinomialPDF(n,p,x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%,B
inomialPDFG6%%\"nG%\"pG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "F:=x->BinomialCDF(n,p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"FGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%,BinomialCDFG6%%\"nG%\"pG9$F(
F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 57 "Below you will find three different ways to calculate P( " }
{XPPEDIT 18 0 "X <=3" "6#1%\"XG\"\"$" }{TEXT -1 6 " ) .  " }}{PARA 0 "
" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"evalf(f(0)+f(1)+f(2)+f(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++
v=<!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(sum(f(x),x=
0..3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++v=<!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(F(3));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+++v=<!#5" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "
Properties of the Distribution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 259 61 "Calculation of Mean, Variance, and M
oment Generating Function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 
0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 14 "Calculating E(" }{TEXT 260 1 "X" }{TEXT -1 43 "), the exp
ectation or mean of the Binomial(" }{TEXT 332 3 "n,p" }{TEXT -1 15 ") \+
distribution." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "f:=x->BinomialPDF(n,p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%,BinomialPDFG6%%\"nG%\"pG9$F(F
(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "EX:=simplify(sum(x*f
(x),x=0..n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EXG,$*&**),$*&\"\"
\"F+,&!\"\"\"\"\"%\"pGF.!\"\"F-%\"nGF.F/F.F1F.),&F.F.F/F-,&F1F.F.F.F.F
+F,F0F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 147 "This is Maple's final simplified form.  However, cancell
ing common factors of the numerator and denominator in the above expre
ssion reveals that E(" }{TEXT 316 1 "X" }{TEXT -1 2 ")=" }{TEXT 317 2 
"np" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Calculati
ng Var(" }{TEXT 261 1 "X" }{TEXT -1 32 "), the variance of the Binomia
l(" }{TEXT 326 3 "n,p" }{TEXT -1 15 ") distribution," }}{PARA 0 "" 0 "
" {TEXT -1 33 "we will employ the formula:  Var(" }{TEXT 262 1 "X" }
{TEXT -1 7 ") = E( " }{XPPEDIT 18 0 "X^2;" "6#*$%\"XG\"\"#" }{TEXT -1 
5 " ) - " }{XPPEDIT 18 0 "( E(X) )^2" "6#*$-%\"EG6#%\"XG\"\"#" }}
{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "E_X_SQ:=simplify(sum(x^2*f(x),x=0..n));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'E_X_SQG*,,(*&%\"nG\"\"\"%\"pGF)F)F)F)F*!\"\"F
)),$*&\"\"\"F/,&F+F)F*F)!\"\"F+F(F)),&F)F)F*F+F(F)F*F/F(F/" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The above
 expression for E( " }{XPPEDIT 18 0 "X^2" "6#*$%\"XG\"\"#" }{TEXT -1 
25 " ) clearly simplifies as " }{XPPEDIT 18 0 "n*p*(n*p+1-p)" "6#*(%\"
nG\"\"\"%\"pGF%,(*&F$F%F&F%F%\"\"\"F%F&!\"\"F%" }{TEXT -1 3 " . " }}
{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "VarX:=simplify(n*p*(n*p+1-p)-(n*p)^2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%VarXG,&*&%\"nG\"\"\"%\"pGF(F(*&)F)\"\"#\"\"\"
F'F-!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 15 "Therefore, Var(" }{TEXT 327 1 "X" }{TEXT -1 17 ") for a B
inomial(" }{TEXT 328 3 "n,p" }{TEXT -1 18 ") distribution is " }
{XPPEDIT 18 0 "n*p*(1-p)" "6#*(%\"nG\"\"\"%\"pGF%,&\"\"\"F%F&!\"\"F%" 
}{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The Momen
t Generating Function (MGF) can be easily calculated via Maple." }}
{PARA 0 "" 0 "" {TEXT -1 59 "Recall the moment generating function of \+
a random variable " }{TEXT 318 1 "X" }{TEXT -1 11 " is defined" }}
{PARA 0 "" 0 "" {TEXT -1 3 "as " }{TEXT 264 1 "M" }{TEXT -1 1 "(" }
{TEXT 265 3 " t " }{TEXT -1 6 ") = E(" }{XPPEDIT 18 0 "exp(tX)" "6#-%$
expG6#%#tXG" }{TEXT -1 36 "), provided this expectation exists." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "simplify(sum(exp(t*x)*f(x),x=0..n));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&),&\"\"\"F&%\"pG!\"\"%\"nGF&),$*&,(F
'F(F&F&*&-%$expG6#%\"tGF&F'F&F&\"\"\",&F(F&F'F&!\"\"F(F)F&" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 145 "This is Maple's final simplified form.  \+
However, cancelling common factors of the numerator and denominator in
 the above expression reveals that " }}{PARA 0 "" 0 "" {TEXT -1 4 "   \+
 " }{TEXT 266 1 "M" }{TEXT -1 2 "( " }{TEXT 267 1 "t" }{TEXT -1 5 " ) \+
= " }{XPPEDIT 18 0 "(1-p+p*exp(t))^n;" "6#),(\"\"\"\"\"\"%\"pG!\"\"*&F
'F&-%$expG6#%\"tGF&F&%\"nG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "We will define " }{TEXT 268 1 "
M" }{TEXT -1 2 "( " }{TEXT 269 1 "t" }{TEXT -1 20 " ) as a function of
 " }{TEXT 270 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "M:=t->
(1-p+p*exp(t))^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGR6#%\"tG6\"
6$%)operatorG%&arrowGF(),(\"\"\"F.%\"pG!\"\"*&F/F.-%$expG6#9$F.F.%\"nG
F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 76 "The moment generating function provides us alternative wa
ys to calculate the" }}{PARA 0 "" 0 "" {TEXT -1 40 "mean and variance \+
by way of the formula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "M^r;" "6#)%\"MG%\"rG" }{TEXT 
-1 10 " (0) = E( " }{XPPEDIT 18 0 "X^r" "6#)%\"XG%\"rG" }{TEXT -1 11 "
 ) , where " }{XPPEDIT 18 0 "M^r" "6#)%\"MG%\"rG" }{TEXT -1 1 "(" }
{TEXT 271 1 "t" }{TEXT -1 14 ") denotes the " }{TEXT 272 1 "r" }{TEXT 
-1 18 " th derivative of " }{TEXT 273 1 "M" }{TEXT -1 1 "(" }{TEXT 
274 1 "t" }{TEXT -1 18 ") with respect to " }{TEXT 275 1 "t" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
30 "This formula holds as long as " }{TEXT 276 1 "M" }{TEXT -1 1 "(" }
{TEXT 277 1 "t" }{TEXT -1 45 ") exists in an open interval containing \+
zero." }}{PARA 0 "" 0 "" {TEXT -1 79 "See, for example, Mathematical S
tatistics and Data Analysis by John A. Rice for" }}{PARA 0 "" 0 "" 
{TEXT -1 39 "more on the moment generating function." }}{PARA 0 "" 0 "
" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M_p:
=diff(M(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M_pG*&**),(%\"pG!
\"\"\"\"\"F+*&-%$expG6#%\"tGF+F)F+F+%\"nGF+F1F+F-\"\"\"F)F2F2F(!\"\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(M_p);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#**),(%\"pG!\"\"\"\"\"F(*&-%$expG6#%\"tGF(F&F
(F(,&%\"nGF(F'F(F(F/F(F*\"\"\"F&F0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "simplify(subs(t=0,M_p));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&%\"nG\"\"\"%\"pGF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "And therefore if " }{TEXT 278 
1 "X" }{TEXT -1 15 " is a Binomial(" }{TEXT 319 3 "n,p" }{TEXT -1 19 "
) variable, then E(" }{TEXT 279 1 "X" }{TEXT -1 5 ") =  " }{TEXT 320 
2 "np" }{TEXT -1 20 " , which agrees with" }}{PARA 0 "" 0 "" {TEXT -1 
57 "what we found earlier.  Now turning to the second moment." }}
{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "M_pp:=diff(M_p,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%%M_ppG,(*&**),(%\"pG!\"\"\"\"\"F,*&-%$expG6#%\"tGF,F*F,F,%\"nGF,)F2
\"\"#\"\"\")F.F4F5)F*F4F5F5*$)F)\"\"#F5!\"\"F,*&**F(F5F2F,F.F5F*F5F5F)
F;F,*&**F(F5F2F5F6F5F7F5F5*$)F)\"\"#F5F;F+" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "simplify(subs(t=0,M_pp));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*&)%\"nG\"\"#\"\"\")%\"pGF'F(\"\"\"*&F&F+F*F+F+*&F)F(
F&F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Therefore
 E( " }{XPPEDIT 18 0 "X^2" "6#*$%\"XG\"\"#" }{TEXT -1 5 " ) = " }
{XPPEDIT 18 0 "n^2*p^2+n*p-n*p^2;" "6#,(*&%\"nG\"\"#%\"pG\"\"#\"\"\"*&
F%F)F'F)F)*&F%F)*$F'\"\"#F)!\"\"" }{TEXT -1 45 " , which again is in a
greement with the value" }}{PARA 0 "" 0 "" {TEXT -1 64 "calculated pre
viously. The variance is now quickly calculated as" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "Var(" }{TEXT 280 1 "X" }{TEXT -1 7 ") = E( " }{XPPEDIT 18 
0 "X^2;" "6#*$%\"XG\"\"#" }{TEXT -1 5 " ) - " }{XPPEDIT 18 0 "( E(X) )
^2" "6#*$-%\"EG6#%\"XG\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 14 "          \+
  = " }{XPPEDIT 18 0 "n^2*p^2+n*p-n*p^2-(n*p)^2;" "6#,**&%\"nG\"\"#%\"
pG\"\"#\"\"\"*&F%F)F'F)F)*&F%F)*$F'\"\"#F)!\"\"*$*&F%F)F'F)\"\"#F." }}
{PARA 0 "" 0 "" {TEXT -1 14 "            = " }{XPPEDIT 18 0 "n*p-n*p^2
;" "6#,&*&%\"nG\"\"\"%\"pGF&F&*&F%F&*$F'\"\"#F&!\"\"" }}{PARA 0 "" 0 "
" {TEXT -1 14 "            = " }{XPPEDIT 18 0 "n*p*(1-p)" "6#*(%\"nG\"
\"\"%\"pGF%,&\"\"\"F%F&!\"\"F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 263 35 "Special Properties of the Binomial(
" }{TEXT 334 3 "n,p" }{TEXT 335 14 ") Distribution" }}{PARA 0 "" 0 "" 
{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 56 "You probably noticed in the previous animations th
at as " }{TEXT 321 1 "n" }{TEXT -1 18 " grew larger with " }{TEXT 333 
1 "p" }{TEXT -1 69 " remaining a fixed value, the probability histogra
m for the Binomial(" }{TEXT 322 3 "n,p" }{TEXT -1 105 ") distribution \+
seemed to become increasingly bell-shaped (i.e., normal). Look again a
t the animation for " }{TEXT 329 1 "n" }{TEXT -1 53 " varying from 4 t
o 50 with an overlayed normal curve." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:=0.2;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"pG$\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "for n from 4 to 50 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 44 "H[n]:=ProbHist(BinomialPDF(n,p,x),0..50,51):" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 49 "N[n]:=plot(NormalPDF(n*p,n*p*(1-p),x),x = 0..50)
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "num:=convert(n, string):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "tracker[n]:=textplot([40,0.4,`n is \+
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y(\{H[n],N[n],tracker[n]\}):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "display([seq(P[n], n=4..50)], \+
insequence=true,title=\"Normal Approximation to the Binomial as n incr
eases\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" 
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\"1fYfw]=96Fcjn7$Fj^m$\"1AI>*GqqC&Fj_qF\\_mF`_mFd_m-%&TITLEG6#QTNormal
~Approximation~to~the~Binomial~as~n~increasesF`_n" 1 2 0 1 0 2 9 1 4 
2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 11 "Indeed, as " }{TEXT 323 1 "n" }{TEXT -1 27 " becom
es larger, for fixed " }{TEXT 324 1 "p" }{TEXT -1 15 ", the Binomial(
" }{TEXT 325 3 "n,p" }{TEXT -1 70 ") distribution becomes increasingly
 better approximated by the Normal(" }{XPPEDIT 18 0 "mu=n*p" "6#/%#muG
*&%\"nG\"\"\"%\"pGF'" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "sigma^2 = n*p*
(1-p)" "6#/*$%&sigmaG\"\"#*(%\"nG\"\"\"%\"pGF),&\"\"\"F)F*!\"\"F)" }
{TEXT -1 16 " ) distribution." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Simulation" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 286 "We will now simul
ate some data from a Binomial distribution in order to compare the sam
pling distribution to the theoretical distribution.  The next bit of c
ode will simulate a random sample of size 500 from a Binomial(10,0.5) \+
distribution, but you are invited to change the parameters." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r
estart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "n:=
10: p:=0.5: numsim:=500:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 
"sample:=BinomialS(n,p,numsim):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 37 "a:=Histogram(sample,-0.5..n+0.5,n+1):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 41 "b:=ProbHist(BinomialPDF(n,p,x),0..n,n+1):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display([a,b],title=\"Simula
ted vs. Theoretical\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6'-%'CURVESG6%7N7$$!1+++++++]!#;\"\"!F'7$$\"1+++++++]F*F+
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$\"1+++++++DF6F97$F<F+F>7$F<$\"1++++++g8F*7$$\"1+++++++NF6F@7$FCF+FE7$
FC$\"1++++++!=#F*7$$\"1+++++++XF6FG7$FJF+FLFI7$$\"1+++++++bF6FG7$FNF+F
P7$FN$\"1++++++S?F*7$$\"1+++++++lF6FR7$FUF+FW7$FU$\"1++++++g6F*7$$\"1+
++++++vF6FY7$FfnF+Fhn7$Ffn$\"1+++++++MF27$$\"1+++++++&)F6Fjn7$F]oF+F_o
7$F]o$\"1+++++++?F27$$\"1+++++++&*F6Fao7$FdoF+FfoFfo7$$\"1++++++]5!#9F
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27$F<FcqF>F>7$F<$\"1++++](=<\"F*7$FCFgqFEFE7$FC$\"1+++]7y]?F*7$FJF[rFL
FL7$FJ$\"1++++v$4Y#F*7$FNF_rFPFP7$FNF[r7$FUF[rFWFW7$FUFgq7$FfnFgqFhnFh
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&F^pF+F+F_p-%+AXESLABELSG6$%!GFas-%&TITLEG6#Q:Simulated~vs.~Theoretica
l6\"-%%VIEWG6$%(DEFAULTGFjs" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Now let's compare \+
the simulated mean to the theoretical mean.  " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "theo_mean:=n*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%*theo_meanG$\"#]!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "sim_mean:=evalf(Mean(sample));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%)sim_meanG$\"++++c\\!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
47 "relative_error:=(sim_mean-theo_mean)/theo_mean;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%/relative_errorG$!+++++))!#7" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 1803 }