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Suppose we now take a sample of size " }{TEXT 281 1 "n" }{TEXT -1 79 " without replacement from this population. The random variable of interest is " }{TEXT 277 1 "X" }{TEXT -1 47 " = the number of good el ements in the sample. " }{TEXT 278 1 "X" }{TEXT -1 79 " follows a hyp ergeometric distribution which will be denoted by Hypergeometric(" } {TEXT 279 7 "G, B, n" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 257 1 "X" }{TEXT -1 48 " is a \+ random variable having the Hypergeometric(" }{TEXT 280 7 "G, B, n" } {TEXT -1 63 ") distribution, then the probability distribution functio n for " }{TEXT 259 1 "X" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " " }{TEXT 260 1 "f" } {TEXT -1 2 " (" }{TEXT 283 1 "x" }{TEXT -1 7 ") = P( " }{TEXT 261 1 "X " }{TEXT -1 3 " = " }{TEXT 284 1 "x" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "binomial(G,x)*binomial(N-G,n-x)/binomial(N,n);" "6#*(-%)binomialG6$ %\"GG%\"xG\"\"\"-F%6$,&%\"NGF)F'!\"\",&%\"nGF)F(F.F)-F%6$F-F0F." } {TEXT -1 2 ", " }{TEXT 285 1 "x" }{TEXT -1 13 " = 1,2, ..., " }{TEXT 286 1 "n" }}{PARA 0 "" 0 "" {TEXT 258 3 " " }}{PARA 0 "" 0 "" {TEXT -1 76 "The following code will draw a probability histogram for the Hy pergeometric(" }{TEXT 282 7 "G, B, n" }{TEXT -1 20 ") distribution wit h " }{TEXT 287 1 "G" }{TEXT -1 5 "=15, " }{TEXT 288 1 "B" }{TEXT -1 8 "=5, and " }{TEXT 289 1 "n" }{TEXT -1 46 "=5. You are encourage to tr y other values of " }{TEXT 290 1 "G" }{TEXT -1 2 ", " }{TEXT 291 1 "B " }{TEXT -1 6 ", and " }{TEXT 292 1 "n" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "G:=15; B:=5; n:=5; " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\" \"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ProbHist(Hypergeomet ricPDF(G,B,n,g),0..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7:7$$!1+++++++]!#;\"\"!7$F($\"1++++%[*\\k!# ?7$$\"1+++++++]F*F-7$F1F+F37$F1$\"1++++8YP[!#=7$$\"1+++++++:!#:F57$F9F +F<7$F9$\"1+++?eWsn!#<7$$\"1+++++++DF;F>7$FBF+FD7$FB$\"1+++AlsMHF*7$$ \"1+++++++NF;FF7$FIF+FK7$FI$\"1+++$y*3-WF*7$$\"1+++++++XF;FM7$FPF+FR7$ FP$\"1+++0&>p$>F*7$$\"1+++++++bF;FT7$FWF+-%'COLOURG6&%$RGBGF+F+$\"*+++ +\"!\")-%+AXESLABELSG6$%!GF^o-%%VIEWG6$%(DEFAULTGFbo" 1 2 0 1 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "To see the effect of changes o f " }{TEXT 293 2 "G " }{TEXT -1 121 "on the shape of the hypergeometri c distribution, the following animation will draw a series of probabil ity histograms as " }{TEXT 294 1 "G" }{TEXT -1 45 " varies from 1 to 2 0 by increments of 1 when " }{TEXT 295 1 "N" }{TEXT -1 8 "=40 and " } {TEXT 296 1 "n" }{TEXT -1 13 "=10. Recall " }{TEXT 297 1 "B" }{TEXT -1 1 "=" }{TEXT 298 4 "N-G." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "B:=40-G: n:=10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for G from 1 to 20 do" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "num:=convert(evalf(G), string) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "tracker[G]:=textplot([8, 0.4, \+ `G is `.num],color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "H[G]:= ProbHist(HypergeometricPDF(G,B,n,x),0..10):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "P[G]:=display(\{H[G],tracker[G]\}):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "display ([seq(P[G], G=1..20)], insequence=true,title=\"N is fixed at 40, G is \+ increasing\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 " 6$-%(ANIMATEG667&-%'CURVESG6$7N7$$!1+++++++]!#;\"\"!7$F,$\"1+++ah%Q:'F .7$$\"1+++++++]F.F17$F4F/F67$F4$\"1+++YQ:YQF.7$$\"1+++++++:!#:F87$F;F/ 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Calculations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 0 "" }{TEXT -1 54 "We can calculate probabilities for the Hypergeo metric(" }{TEXT 302 7 "G, B, n" }{TEXT -1 43 ") distribution using eit her the probability" }}{PARA 0 "" 0 "" {TEXT -1 90 "distribution funct ion (PDF) or the cumulative distribution function (CDF). We will deno te" }}{PARA 0 "" 0 "" {TEXT -1 11 "the PDF by " }{TEXT 263 1 "f" } {TEXT -1 16 " and the CDF by " }{TEXT 264 1 "F" }{TEXT -1 51 ". We wi ll consider these functions for a variable " }{TEXT 269 1 "X" }{TEXT -1 7 " having" }}{PARA 0 "" 0 "" {TEXT -1 39 "the Geometric(30, 10, 10 ) distribution." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "G:=30; B:=10; n:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG\"# 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "f:=t->HypergeometricPDF(G,B,n,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%2Hype rgeometricPDFG6&%\"GG%\"BG%\"nG9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "At the present time , there is no HypergeometricCDF command. We can create our own by sum ming the HypergeometricPDF." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 23 "F:=x->sum(f(t), t=0..x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6#%\"xG6\"6$%)opera torG%&arrowGF(-%$sumG6$-%\"fG6#%\"tG/F2;\"\"!9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Below you will find three different ways to calculate P( " }{XPPEDIT 18 0 "X <= 3" "6#1%\"XG\"\"$" }{TEXT -1 6 " ) . " }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(f(1)+f(2)+f (3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+fF0#)f!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(sum(f(x),x=1..3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+fF0#)f!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(F(3)-F(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+fF0#)f!#8" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Properti es of the Distribution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 265 32 "Calculation of Mean and Variance" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Calculating E(" } {TEXT 266 1 "X" }{TEXT -1 49 "), the expectation or mean of the hyperg eometric(" }{TEXT 303 7 "G, B, n" }{TEXT -1 15 ") distribution." }} {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f(x):=Hype rgeometricPDF(G,B,n,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#% \"xG*&*&-%)binomialG6$%\"GGF'\"\"\"-F+6$%\"BG,&%\"nGF.F'!\"\"F.\"\"\"- F+6$,&F-F.F1F.F3!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "EX :=simplify(sum(x*f(x),x=0..G));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# EXG*&*&%\"GG\"\"\"%\"nGF(\"\"\",&F'F(%\"BGF(!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Calculating Var(" }{TEXT 267 1 "X" }{TEXT -1 38 "), the variance of the Hypergeometric( " }{TEXT 304 7 "G, B, n" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 33 "We will employ the formula: Var(" }{TEXT 268 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 36 "EXX:=s implify(sum(x^2*f(x),x=0..G));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E XXG*&*(%\"GG\"\"\"%\"nGF(,(%\"BG\"\"\"*&F'F,F)F,F,F)!\"\"F,F(*&,(F'F,F +F,F.F,\"\"\",&F'F,F+F,\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "VarX:=simplify(EXX-EX^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VarXG*&**%\"GG\"\"\"%\"nGF(%\"BGF(,(F'F(F*F(F)!\"\"F (\"\"\"*&,(F'F(F*F(F,F(\"\"\"),&F'F(F*F(\"\"#F-!\"\"" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Simulation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 232 "We will now simulate som e data from a hypergeometric distribution in order to compare the samp ling distribution to the theoretical distribution. The next bit of co de will simulate a random sample of size 500 from a Hypergeometric (" }{TEXT 271 1 "G" }{TEXT -1 7 " = 50, " }{TEXT 272 2 "B " }{TEXT -1 6 " = 25, " }{TEXT 305 1 "n" }{TEXT -1 66 " = 30) distribution, but you ar e invited to change the parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "G:=50: B:=25: n:=30: numsim: =500:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sample:=HypergeometricS(G, B, n,numsim):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "a:=Histogram(sample,-0.5..30 +0.5,31):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "b:=ProbHist(Hy pergeometricPDF(G,B,n,g),0..30,30):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display([a,b],title=\"Simulated(red) vs. Theoretical( blue)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%' CURVESG6%7hr7$$!1+++++++]!#;\"\"!F'7$$\"1+++++++]F*F+F,F,F,7$$\"1+++++ ++:!#:F+F/F/F/7$$\"1+++++++DF2F+F3F3F37$$\"1+++++++NF2F+F6F6F67$$\"1++ +++++XF2F+F9F9F97$$\"1+++++++bF2F+FFNFhp7$F[qF+F]q7$F[q$\"1++++++g=F*7$$\"1++++++]?FNF_ q7$FbqF+Fdq7$Fbq$\"1++++++!o\"F*7$$\"1++++++]@FNFfq7$FiqF+F[r7$Fiq$\"1 ++++++g7F*7$$\"1++++++]AFNF]r7$F`rF+Fbr7$F`r$\"1+++++++iF]o7$$\"1+++++ +]BFNFdr7$FgrF+Fir7$Fgr$\"1+++++++AF]o7$$\"1++++++]CFNF[s7$F^sF+F`s7$F ^s$FZF]o7$$\"1++++++]DFNFbs7$FdsF+FfsFfs7$$\"1++++++]EFNF+FgsFgsFgs7$$ \"1++++++]FFNF+FjsFjsFjs7$$\"1++++++]GFNF+F]tF]tF]t7$$\"1++++++]HFNF+F `tF`tF`t7$$\"1++++++]IFNF+Fct-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*LI NESTYLEG6#\"\"$-F$6$7hrF'F'F,F,F,F,F/F/F/F/F3F3F3F3F6F6F6F6F9F9F97$F:$ \"1+++o&H)4F!#I7$F=FeuF7$FVF]xFUFU7$FV$ \"1+++(f*\\]CFen7$FgnFbxFinFin7$Fgn$\"1+++_%=*4%*Fen7$F_oFfxFaoFao7$F_ o$\"1+++-f$p!GF]o7$FfoFjxFhoFho7$Ffo$\"1+++0r^\\lF]o7$F]pF^yF_pF_p7$F] p$\"1+++-[u+7F*7$FdpFbyFfpFfp7$Fdp$\"1+++IgSLF*7$FbqFjyFdqFdq7$Fbq$\"1+++j2?fFen7$FhsFb[lFgsFgs7$Fhs$\"1+++pFByJF_x7$F[tFf[lFjsFjs7$F[t$\"1+ ++&R\\_S$Fjw7$F^tFj[lF]tF]t7$F^t$\"1+++rQu_@Few7$FatF^\\lF`tF`t7$Fat$ \"1+++OGoFgF[w7$FdtFb\\lFct-Fgt6&FitF+F+Fjt-%+AXESLABELSG6$%!GFj\\l-%& TITLEG6#QESimulated(red)~vs.~Theoretical(blue)6\"-%%VIEWG6$%(DEFAULTGF c]l" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Now let's compare the simulated mean to t he theoretical mean." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "theo_mean:=n*G/(G+B);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%*theo_meanG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sim_mean:=evalf(Mean(sample));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)sim_meanG$\"+++S(*>!\")" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "relative_error:=(sim_mean-theo_mean)/theo_mean;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%/relative_errorG$!+++++8!#7" }}}}} {MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 }