{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "Verdana" 1 24 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE " " -1 257 "Arial Narrow" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "cmbx12" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Verdana " 1 24 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Times" 0 14 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Verdana" 1 24 0 0 0 0 0 0 2 0 0 0 0 0 0 } {CSTYLE "" -1 271 "Helvetica" 1 24 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "dcbxsl10" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 259 1 {CSTYLE "" -1 -1 " Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "" 257 262 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 257 263 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 257 264 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 257 265 1 {CSTYLE "" -1 -1 "Helveti ca" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 257 267 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }}{PARA 18 "" 0 "" {TEXT 256 17 "Using M aple for T" }{TEXT 271 0 "" }{TEXT 270 8 "eaching " }}{PARA 18 "" 0 " " {TEXT 259 26 "Probability and Statistics" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 15 "Zaven A. Karian" }}{PARA 263 "" 0 "" {TEXT -1 15 "Department of M " }{TEXT 258 0 "" }{TEXT -1 31 "athematics and Computer Science" }} {PARA 264 "" 0 "" {TEXT -1 18 "Denison University" }}{PARA 265 "" 0 " " {TEXT -1 21 "Granville, OH 43023" }}{PARA 266 "" 0 "" {TEXT -1 0 " " }}{PARA 267 "" 0 "" {TEXT -1 18 "Karian@Denison.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 260 60 "1. Computational Support for Probability and Statistics" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Indepen dently of the statistics package of Maple, we " }}{PARA 0 "" 0 "" {TEXT -1 62 "have developed a statistical supplement that cons ists of" }}{PARA 0 "" 0 "" {TEXT -1 39 "about 140 routines. These \+ routines " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "1.1. Perform basic computations for descriptive statistics" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "1.2. Gene rate random samples from known discrete and" }}{PARA 0 "" 0 "" {TEXT -1 62 " continuous distributions as well as from distributions \+ " }}{PARA 0 "" 0 "" {TEXT -1 30 " with specified p.d.f.'s." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "1.3. Plot histograms, empirical p.d.f.s and c.d.f.s, " }}{PARA 0 "" 0 "" {TEXT -1 49 " box-and-whisker diagrams, q-q graphs, etc." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "1.4. Perform linea r and polynomial regression fits and produce" }}{PARA 0 "" 0 "" {TEXT -1 49 " graphic displays associated with such fits." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "1.5. Produce the p .d.f.s, c.d.f.s, and percentiles of most " }}{PARA 0 "" 0 "" {TEXT -1 41 " commonly encountered distributions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "1.6. Generate samples fro m certain sampling distributions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 58 "1.7. Do computations and graphic displays associated with " }}{PARA 0 "" 0 "" {TEXT -1 25 " confidence inter vals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 " 1.8. Perform one-way and two-way analysis of variance " }}{PARA 0 "" 0 "" {TEXT -1 17 " computations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The expl orations and investigations about to be described" }}{PARA 0 "" 0 "" {TEXT -1 16 "are taken from: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 33 " Z. A. Karian & E. A. Tanis, " }} {PARA 0 "" 0 "" {TEXT -1 59 " Probability and Statistics Explo rations with Maple" }}{PARA 0 "" 0 "" {TEXT -1 22 " Prentice-H all" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 56 "The statistical supplement that will be \+ illustrated here" }}{PARA 0 "" 0 "" {TEXT -1 37 "can be obtained via a nonymous FTP by " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "1. connecting to www.denison.edu, " }}{PARA 0 "" 0 "" {TEXT -1 46 "2. moving to the subdirectory pub/mathsci/stat" }}{PARA 0 "" 0 "" {TEXT -1 33 "3. obtaining the text file README" }}{PARA 0 " " 0 "" {TEXT -1 40 "4. following the instructions in README." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 261 34 "2. About the Statistic s Package" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 " 2.1. Simple Compu tations and Plotting" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 24 "wi th(plots): with(stat);" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 30 "B. Generate 15 random numbers" }}{PARA 260 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 13 "X : = RNG(15);" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 34 "C. Compute some sample statistics" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "X := RNG(50) :" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 16 "xbar := Mean(X);" }} }{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 14 "s := StDev(X);" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 27 "median:=Percentile(X, 0.5) ;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 32 "pi[0.25] := Percenti le(X, 0.25);" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 34 "D. Obtain a histogram of the data" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 23 "Histogram(X, 0..1, 10);" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 29 "E. Plot the running averages " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "PlotRunningAverage(X);" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 32 "F. Do this for a larger sam ple" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 14 "A := RNG(500):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 27 "R := PlotRunningAverage(A):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 23 "P := plot(0.5, 0..500):" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 18 "display( \{R, P\} );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 " 2.2. The Help System " }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 6 "? Mean" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 11 "? Histogram" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 262 23 "3. Simple Simulation s" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 " 3.1. Rolling Dice" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 31 "A. Rol l a 6-sided die 10 times" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 15 "A := Die(6,10);" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 " " 0 "" {TEXT -1 42 "B. Obtain the frequencies of the outcomes" } {MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 14 "Freq(A , 1..6);" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 43 "C. Repeat with 600 rolls of a 12-sided die" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 18 "B := Die(12, 600) :" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 20 "F := Freq(B, 1..12); " }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 39 " D. Obtain a histogram of the 600 rolls" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 28 "Histogram(B, 0.5..12.5, 12);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 " 3.2. Poker Hands" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 55 "A. Loa d the randcomb routine from the combinat package" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 25 "with(combi nat, randcomb):" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 " " {TEXT -1 55 "B. Cards is a data structure defined by the supplement " }}{PARA 260 "" 0 "" {TEXT -1 38 " use it to generate a poker h and" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 171 "Cards := [c2,c3,c4,c5,c6,c7,c8,c 9,c10,cJ,cQ,cK,cA,d2,d3,d4,d5,d6,d7,d8,d9,d10,dJ,dQ,dK,dA,h2,h3,h4,h5, h6,h7,h8,h9,h10,hJ,hQ,hK,hA,s2,s3,s4,s5,s6,s7,s8,s9,s10,sJ,sQ,sK,sA]; " }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 32 "PokerHand := randcomb (Cards, 5);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 43 "C. Use it again to generate 10 poker hands" }} {PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 51 "PokerHands := [seq(randcomb(Cards, 5), \+ i = 1..10)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 " 3.3. \+ Flipping coins" }}{PARA 259 "" 0 "" {TEXT 263 57 "A. Coin flips can \+ be simulated by using the Die function" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 16 "F := Die(2, 10);" } }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 32 "Coins := [seq(F[i]-1, i= 1..10)];" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 55 "B. Or, by generating a sample from the Bernoulli dist." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 29 "Coins := BernoulliS(0.5, 10);" }}} {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 62 "C. O r, by generating a sample from the discrete uniform dist." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 32 "Coins := DiscUniformS(0..1, 10);" }}}{PARA 260 " " 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 40 "D. Obtain 3 set s of 600 flips of a coin" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 33 "Flip1 := DiscUn iformS(0..1, 600);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 32 "Fli p2 := DiscUniformS(0..1,600):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 33 "Flip3 := DiscUniformS(0..1, 600):" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 53 "E. Obtain the number of head s in each set of 3 flips" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 66 "NumberHeads := \+ [seq( Flip1[i] + Flip2[i] + Flip3[i], i = 1..600)]:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 37 "FreqHeads := Freq(NumberHeads, 0..3);" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 48 "F. Look at the histogram of the number of heads" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 37 "Histogram(NumberHeads, -0.5..3.5, 4);" }}}{PARA 260 " " 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 50 "G. Generate the number of heads in 16 flips using" }}{PARA 260 "" 0 "" {TEXT -1 35 " \+ the binomial distribution " }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 34 "Sampl e := BinomialS(16, 1/2, 600);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "Freq(Sample, 0..16);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 34 "Histogram(Sample, -0.5..16.5, 17);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 264 48 "4. Investigating Properties of Distributions" }{TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 37 " ANIMATION TO STUDY" }}{PARA 260 "" 0 "" {TEXT -1 6 " " }}{PARA 260 "" 0 "" {TEXT -1 38 " FAMILIES OF DIS TRIBUTIONS" }}{PARA 260 "" 0 "" {TEXT -1 5 " " }}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 25 " 4.1. The t-Distribution" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "with(plots, animate): " }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 51 "A. C ompare the t-distribution with df=1 to N(0,1)." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 16 "T := TPDF(1, x);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "N := NormalPDF(0,1,x);" }}}{EXCHG {PARA 260 "> " 0 " " {MPLTEXT 1 0 28 "plot(\{N,T\}, x= -4.5 .. 4.5);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 54 "B. It is m ore effective to consider the t-dist. with" }}{PARA 260 "" 0 "" {TEXT -1 49 " df=1, 2, ..., 16 and look at an animation." }} {PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 17 "T := TPDF(nu, x);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 55 "animate(\{T,N\}, x=-4.5 .. 4.5, nu=1 .. 16 , color=blue);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 53 "C. What is \"odd\" about the t-distribution when df=1?" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 15 "A := TS(1, 20);" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "Mean(A); Variance(A);" }} }{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "PlotRunningAverage(A);" } }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "B := NormalS(0,1, 20):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "Mean(B); Variance(B); " }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 16 "A := TS(1, 200):" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "Mean(A); Variance(A);" }} }{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "PlotRunningAverage(A);" } }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 31 "A[8..12]; A[23..27]; A[3 4..38];" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 17 "Max(A); Min( A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 34 " 4.2. The Chi-Square Distribution" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "with(plots, animate):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 15 "assume(nu > 0);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 25 "f := ChisquarePDF(nu, x);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 42 "mu := simplify(int(x*f, x = 0..in finity));" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 50 "Var := simpl ify(int(f*x^2, x = 0..infinity)-mu^2);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 45 "animate(f, x = 1..25, nu = 1..5, color=blue);" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 46 "plot3d(f, x = 0..10, nu = \+ 1..5, axes = boxed);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 29 "d erf := simplify(diff(f, x));" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 19 "solve(derf = 0, x);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 32 " 4.3. The Binomial \+ Distribution" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 33 " THE BINOMIAL DISTRIBUTION AND" }}{PARA 260 "" 0 "" {TEXT -1 31 " ITS APPROXIMATIONS" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "with(plots , display):" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 50 "A. Generate a sample from a binomial distribution " }}{PARA 260 "" 0 "" {TEXT -1 53 " and compare empirical and theoret ical hsitograms " }}{PARA 260 "" 0 "" {TEXT -1 55 " and empirical a nd theoretic distribution functions." }}{PARA 260 "" 0 "" {TEXT -1 0 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 29 "A \+ := BinomialS(10, 0.7, 200);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 29 "Histogram(A, -0.5..10.5, 11);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 31 "pdf := BinomialPDF(10, 0.7, x);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "ProbHist(pdf, 0..10);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 34 "H := Histogram(A, -0.5..10.5, 11):" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 26 "P := ProbHist(pdf, 0..10): " }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 17 "display( \{H,P\} );" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 28 "Emp := PlotEmpCDF(A, 0. .10):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "Th := PlotDiscCD F(pdf, 0..10):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "display ( \{Emp, Th\} );" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 51 "B. We can look at the binomial p.d.f. symboli cally " }}{PARA 260 "" 0 "" {TEXT -1 55 " and obtain certain proper ties of this distribution." }}{PARA 260 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 26 "y := \+ BinomialPDF(n, p, x);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 17 " sum(y, x = 0..n);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "simp lify(%, symbolic);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 23 "mu: =sum(x*y, x = 0..n);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 26 "m u:=simplify(%, symbolic);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" } }{PARA 260 "" 0 "" {TEXT -1 51 "C. Look at the shape of the binimial f or a fixed p " }}{PARA 260 "" 0 "" {TEXT -1 33 " (p = 0.25) and 2 \+ <= n <= 32." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> \+ " 0 "" {MPLTEXT 1 0 21 "for i from 1 to 16 do" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 62 " H.i := ProbHist(BinomialPDF(2*i, 0.25, x), -0.5..1 6.5, 17):" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 48 "display([seq(H.i, i = 1..16)], insequen ce=true);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 48 "D. Is N(np, npq) a good approximation of b(n,p)?" }} {PARA 260 "" 0 "" {TEXT -1 34 " If so, for what values of n ?" }} {PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to 16 do" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 59 " H := ProbHist(BinomialPDF(2*i, 0.25, x), -0.5..16.5 , 17):" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 54 " N := plot(NormalPDF(i /2, 3*i/8, x), x = -0.5..16.5):" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 25 " P.i := display( \{N,H\} ):" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 48 "display([seq(P.i, i = 1..16)], insequence=true);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 273 11 "5. Underst" }{TEXT -1 0 "" }{TEXT 272 30 "anding an Importa nt Theorem" }}{PARA 261 "" 0 "" {TEXT -1 8 " " }{TEXT 265 28 " THE CENTRAL LIMIT THEOREM" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }} {PARA 260 "" 0 "" {TEXT -1 39 "1. STUDY AN \"ARBITRARY\" DISRIBUTI ON" }}{PARA 260 "" 0 "" {TEXT -1 38 "2. GENERATE RANDOM SAMPLES FRO M IT" }}{PARA 260 "" 0 "" {TEXT -1 36 "3. ILLUSTRATE THE CENTRAL \+ LIMIT " }}{PARA 260 "" 0 "" {TEXT -1 23 " THEOREM USING IT." }} {PARA 260 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "restart: with(plots, display):" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 26 "A. First define the p.d .f." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> (3/2)*x^2;" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 " " }}{PARA 260 "" 0 "" {TEXT -1 42 "B. Do some simple computations (e.g ., the " }}{PARA 260 "" 0 "" {TEXT -1 26 " mean, variance, etc.)" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 19 "int(f(x), x=-1..1);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 27 "mu := int(x*f(x), x=-1..1);" }}}{EXCHG {PARA 260 "> \+ " 0 "" {MPLTEXT 1 0 37 "var := int(x^2*f(x), x=-1..1) - mu^2;" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 24 "F := int(f(t), t=-1..x);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 44 "C. Generate a small random sample (size 10) " }}{PARA 260 "" 0 "" {TEXT -1 28 " from this distribution." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 34 "A := Cont inuousS(f(x), -1..1, 10);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" } }{PARA 260 "" 0 "" {TEXT -1 38 "D. Now produce a larger random sample \+ " }}{PARA 260 "" 0 "" {TEXT -1 41 " (size 600) and compare its hist ogram " }}{PARA 260 "" 0 "" {TEXT -1 17 " to the p.d.f." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 35 "A := ContinuousS(f(x), -1..1, 600):" }}}{EXCHG {PARA 260 "> " 0 " " {MPLTEXT 1 0 29 "H := Histogram(A, -1..1, 12):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 25 "P := plot(f(x), x=-1..1):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 15 "display(\{P,H\});" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 49 "E. To illustrat e the CLT, first consider samples " }}{PARA 260 "" 0 "" {TEXT -1 13 " \+ of size 4" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 50 "A := [seq(ContinuousS(f(x), -1..1, 4), i=1..1 00)]:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 5 "A[1];" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 33 "M := [seq(Mean(A[i]), i=1..100)]: " }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 29 "H := Histogram(M, -1. .1, 12):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 28 "n := NormalPD F(0, var/4, x):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "N := p lot(n, x=-1..1):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 17 "displ ay(\{P,H,N\});" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 41 "F. See if sample size of 8 will be better" }} {PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 50 "A := [seq(ContinuousS(f(x), -1..1, 8), i=1..100)]:" } }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 5 "A[1];" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 33 "M := [seq(Mean(A[i]), i=1..100)]:" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 29 "H := Histogram(M, -1..1, 1 0):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 28 "n := NormalPDF(0, \+ var/8, x):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "N := plot(n , x=-1..1):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 17 "display(\{ P,H,N\});" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 30 "G. Next try samples of size 16" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 51 "A := [seq (ContinuousS(f(x), -1..1, 16), i=1..100)]:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 5 "A[1];" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 33 "M := [seq(Mean(A[i]), i=1..100)]:" }}}{EXCHG {PARA 260 "> " 0 " " {MPLTEXT 1 0 32 "H := Histogram(M, -0.6..0.6, 8):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 29 "n := NormalPDF(0, var/16, x):" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 26 "N := plot(n, x=-0.6..0.6): " }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 17 "display(\{P,H,N\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 275 45 "6. Relationships Between Random Variables" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 61 "A. Illustration of: I f Z is N(0,1) then Z^2 is Chisquare(1)" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 46 " First start with 500 N(0,1) observations." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 " > " 0 "" {MPLTEXT 1 0 23 "Z1 := NormalS(0,1,500):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 31 "Z1H := Histogram(Z1, -4..4, 9):" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 39 "N01 := plot(NormalPDF(0,1, x), x=-4..4):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 20 "display( \{Z1H, N01\});" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 45 " Now obtain Y = Z1^2 and compare this to " }} {PARA 260 "" 0 "" {TEXT -1 18 " Chisquare(1)." }}{PARA 260 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "Y := [s eq(Z1[i]^2, i=1..500)]:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "YH := Histogram(Y, 0..12, 16):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "CH1 := plot(ChisquarePDF(1,x), x=0..12):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 19 "display(\{YH, CH1\});" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 45 "B. N ow consider Y = Z1^2 + Z2^2 + Z3^2 where" }}{PARA 260 "" 0 "" {TEXT -1 50 " Z1, Z2, Z3 are independent N(0,1) and compare" }}{PARA 260 "" 0 "" {TEXT -1 23 " Y to Chisquare(3)." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 23 "Z2 := Nor malS(0,1,500):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 23 "Z3 := N ormalS(0,1,500):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 46 "Y := \+ [seq(Z1[i]^2+Z2[i]^2+Z3[i]^2, i=1..500)]:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "YH := Histogram(Y, 0..18, 24):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "CH3 := plot(ChisquarePDF(3,x), x=0..12) :" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 19 "display(\{YH, CH3\}) ;" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 25 "C. Let X1, ... Xn be N( " }{XPPEDIT 18 0 "mu;" "6#%#muG " }{TEXT -1 1 "," }{XPPEDIT 18 0 "sigma^2;" "6#*$%&sigmaG\"\"#" } {TEXT -1 33 "); (for simplicity, we could take" }}{PARA 260 "" 0 "" {TEXT -1 54 " X1, ... Xn to be N(0,1) in the example below). " } }{PARA 260 "" 0 "" {TEXT -1 13 " If Let " }{XPPEDIT 18 0 "S = Sum ((Z[i]-mu)^2/(sigma^2),i = 1 .. n);" "6#/%\"SG-%$SumG6$*&,&&%\"ZG6#%\" iG\"\"\"%#muG!\"\"\"\"#*$%&sigmaG\"\"#F0/F-;\"\"\"%\"nG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "T = Sum((Z[i]-mean(Z[i]))^2/(sigma^2),i = 1 .. n);" "6#/%\"TG-%$SumG6$*&,&&%\"ZG6#%\"iG\"\"\"-%%meanG6#&F+6#F-!\"\" \"\"#*$%&sigmaG\"\"#F4/F-;\"\"\"%\"nG" }{TEXT -1 3 ". " }}{PARA 260 " " 0 "" {TEXT -1 20 " Show that S is " }{XPPEDIT 18 0 "chi^2;" "6#* $%$chiG\"\"#" }{TEXT -1 13 "(n) and T is " }{XPPEDIT 18 0 "chi^2;" "6# *$%$chiG\"\"#" }{TEXT -1 6 "(n-1)." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 56 " \+ We start by letting n=3, and X1, X2, X3 be N(37,16)" }}{PARA 260 "" 0 "" {TEXT -1 53 " and we take 200 such samples. Next, we compute " }}{PARA 260 "" 0 "" {TEXT -1 58 " S and T and compare them to t he appropriate chisquare" }}{PARA 260 "" 0 "" {TEXT -1 19 " distri butions." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 41 "X := [seq(NormalS(37, 16, 3), i=1..200)]:" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 54 "S := [seq(sum( (X[i][j]-37 )^2/16, j=1..3), i=1..200)]:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 62 "T := [seq(sum( (X[i][j]-Mean(X[i]))^2/16, j=1..3), i=1..200)]: " }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "SH := Histogram(S, 0. .20, 16):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "TH := Histog ram(T, 0..20, 16):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "CH2 := plot(ChisquarePDF(2,x), x=0..20):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "CH3 := plot(ChisquarePDF(3,x), x=0..20):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 19 "display(\{CH3, SH\});" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 19 "display(\{CH2, TH\});" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 276 25 "7. Confide nce Intervals" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 52 "Given a random sample from N(mu, sigma^2), how does " }} {PARA 260 "" 0 "" {TEXT -1 55 "knowledge of sigma^2 affect confidence \+ intervals of mu?" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 " " {TEXT -1 55 "First we extract 25 random samples, each of size 5 from " }}{PARA 260 "" 0 "" {TEXT -1 58 "N(40, 12). Next, we graphically dis play the 80% confidence" }}{PARA 260 "" 0 "" {TEXT -1 13 "intervals of " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 7 " using " }{XPPEDIT 18 0 "s^2;" "6#*$%\"sG\"\"#" }{TEXT -1 24 " as an approximation to " } {XPPEDIT 18 0 "sigma^2;" "6#*$%&sigmaG\"\"#" }{TEXT -1 11 ". Finally, " }}{PARA 260 "" 0 "" {TEXT -1 57 "we plot the confidence intervals us ing the knowledge that" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "sigma^2 = 1 2;" "6#/*$%&sigmaG\"\"#\"#7" }{TEXT -1 1 "." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 20 "with(plots ,display):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 50 "ListOfSampl es := [seq(NormalS(40,12,5), i=1..25)]:" }}}{EXCHG {PARA 260 "> " 0 " " {MPLTEXT 1 0 47 "CIVarUnknown := ConfIntMean(ListOfSamples, 80):" }} }{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "ConfIntPlot(CIVarUnknown, 40);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 49 "CIVarKnown := Co nfIntMean(ListOfSamples, 80, 12):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 28 "ConfIntPlot(CIVarKnown, 40);" }}}{EXCHG {PARA 260 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 266 46 " 8. Power Function of a Statistical Test" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 37 " A Study of Power Functions " }}{PARA 260 "" 0 "" {TEXT -1 39 " Associated with the testing " }}{PARA 260 "" 0 "" {TEXT -1 34 " of the Variance" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 54 "Consider the following statistical testing situation: \+ " }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 63 "X is a normal random variable with variance V. We wish to test " }} {PARA 260 "" 0 "" {TEXT -1 38 " H0: V = 100 against the alternative " }}{PARA 260 "" 0 "" {TEXT -1 18 " H1: V < > 100. " }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "with( plots, display):" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 44 "A. First we fix the sample size n at 21 and \+ " }}{PARA 260 "" 0 "" {TEXT -1 16 " vary alpha." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 19 "n := 2 1; var:=100;" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 " " 0 "" {TEXT -1 54 "B. Specifically for alpha=0.05, we calculate L and U, " }}{PARA 260 "" 0 "" {TEXT -1 53 " the left and right end-poi nts of the confidence " }}{PARA 260 "" 0 "" {TEXT -1 40 " interval associated with this test." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 12 "alpha:=0.05;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 42 "L := var*ChisquareP(n-1, alpha/2) /sigma^2;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 44 "U := var*Chi squareP(n-1, 1-alpha/2)/sigma^2;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 52 "K := 1-ChisquareCDF(n-1,U) + ChisquareCDF(n-1,L); " }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "plot(K, sigma=0..20); " }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 71 "C. We repeat this process for alpha = 0.005, 0.01, 0.025, 0.1, \+ and 0.2." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 13 "alpha:=0.005;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "L:=var*ChisquareP(n-1, alpha/2)/sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 42 "U:=var*ChisquareP(n-1, 1-alpha/2) /sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 49 "K:=1-Chisqua reCDF(n-1,U) + ChisquareCDF(n-1,L):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 26 "P.1:=plot(K, sigma=0..20):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 12 "alpha:=0.01;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "L:=var*ChisquareP(n-1, alpha/2)/sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 42 "U:=var*ChisquareP(n-1, 1-alpha/2) /sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 45 "K:=1-Chisqua reCDF(n-1,U)+ChisquareCDF(n-1,L):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 26 "P.2:=plot(K, sigma=0..20):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 13 "alpha:=0.025;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "L:=var*ChisquareP(n-1, alpha/2)/sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 42 "U:=var*ChisquareP(n-1, 1-alpha/2) /sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 45 "K:=1-Chisqua reCDF(n-1,U)+ChisquareCDF(n-1,L):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 26 "P.3:=plot(K, sigma=0..20):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 12 "alpha:=0.05;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "L:=var*ChisquareP(n-1, alpha/2)/sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 42 "U:=var*ChisquareP(n-1, 1-alpha/2) /sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 45 "K:=1-Chisqua reCDF(n-1,U)+ChisquareCDF(n-1,L):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 26 "P.4:=plot(K, sigma=0..20):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 11 "alpha:=0.1;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "L:=var*ChisquareP(n-1, alpha/2)/sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 42 "U:=var*ChisquareP(n-1, 1-alpha/2) /sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 45 "K:=1-Chisqua reCDF(n-1,U)+ChisquareCDF(n-1,L):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 26 "P.5:=plot(K, sigma=0..20):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 11 "alpha:=0.2;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "L:=var*ChisquareP(n-1, alpha/2)/sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 42 "U:=var*ChisquareP(n-1, 1-alpha/2) /sigma^2:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 45 "K:=1-Chisqua reCDF(n-1,U)+ChisquareCDF(n-1,L):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 26 "P.6:=plot(K, sigma=0..20):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 46 "display([P1,P2,P3,P4,P5,P6], insequence=true);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 29 "display(\{P1,P2,P3,P4,P 5,P6\});" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 " " {TEXT -1 40 "D. We can also study the power function " }}{PARA 260 " " 0 "" {TEXT -1 47 " when alpha is fixed at 0.05 and n assumes " } }{PARA 260 "" 0 "" {TEXT -1 49 " a variety of values (n=5, 9, 13, \+ 17, 21, 25 " }}{PARA 260 "" 0 "" {TEXT -1 23 " in the following). " }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 12 "alpha:=0.05;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to 6 do" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 16 " \+ n := 1+ 4*i:" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 44 " L:=var*Chi squareP(n-1, alpha/2)/sigma^2:" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 46 " U:=var*ChisquareP(n-1, 1-alpha/2)/sigma^2:" }}{PARA 260 "> " 0 " " {MPLTEXT 1 0 49 " K:=1-ChisquareCDF(n-1,U)+ChisquareCDF(n-1,L):" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 30 " Q.i:=plot(K, sigma=0..20): " }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 51 "display([Q1, Q2, Q3, Q4, Q5, Q6], insequence=tru e);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 34 "display(\{Q1, Q2, \+ Q3, Q4, Q5, Q6\});" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 267 41 " 9. Minimum-Length Confidence Intervals" }{TEXT -1 0 "" }} {EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 55 "Minimal length CI for variance based on a random sample" }}{PARA 260 "" 0 "" {TEXT -1 7 "from N(" }{XPPEDIT 18 0 "mu;" "6#%#muG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "sigma^2;" "6#*$%&sigmaG\"\"#" }{TEXT -1 2 ")." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 " " {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 65 "A. Using the ordianr y CI (with equal probability in each tail), " }}{PARA 260 "" 0 "" {TEXT -1 63 " we get the following for the length of the 95% CI fo r the " }}{PARA 260 "" 0 "" {TEXT -1 23 " variance when n=7." }} {PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "n := 7: alpha := 0.05: S:='S':" }}}{EXCHG {PARA 260 " > " 0 "" {MPLTEXT 1 0 80 "UsualL := S^2*(n-1)*( 1/ChisquareP(n-1,alpha /2) - 1/ChisquareP(n-1,1-alpha/2) );" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 57 "B. An optimal length C I can be obtained by defining and " }}{PARA 260 "" 0 "" {TEXT -1 63 " \+ solving two equations (eq1, and eq2 below) simultaneously." }} {PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 23 "f:=ChisquarePDF(n-1,x);" }}}{EXCHG {PARA 260 "> " 0 " " {MPLTEXT 1 0 30 "eq1 := int(f, x=a..b)=1-alpha;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 37 "eq2:=a^2*subs(x=a,f)=b^2*subs(x=b,f);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 64 "solution := fsolve(\{eq 1,eq2\},\{a,b\},\{a=0..n-1, b=n-1..infinity\});" }}}{EXCHG {PARA 260 " > " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 32 "MinimumL := S^2*(n-1)*(1/a-1/b);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 29 "100*(UsualL-MinimumL)/UsualL;" }}} {EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 53 "C. Continuing with n=7, we can compare the usual and" }}{PARA 260 "" 0 "" {TEXT -1 51 " minimum lenghts of CIs for various valu es of " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 6 " e.g.," }} {PARA 260 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "alpha = (.10, .15, .20 , .25, .30, .35, .40, .45, .50, .55, .60, .65, .70, .75, .80, .85, .90 , .95, 1.00, 1.05, 1.10, 1.15, 1.20, 1.25, 1.30, 1.35, 1.40, 1.45, 1.5 0, 1.55, 1.60, 1.65, 1.70, 1.75, 1.80, 1.85, 1.90, 1.95, 2.00);" "6#/% &alphaG6I$\"#5!\"#$\"#:!\"#$\"#?!\"#$\"#D!\"#$\"#I!\"#$\"#N!\"#$\"#S! \"#$\"#X!\"#$\"#]!\"#$\"#b!\"#$\"#g!\"#$\"#l!\"#$\"#q!\"#$\"#v!\"#$\"# !)!\"#$\"#&)!\"#$\"#!*!\"#$\"#&*!\"#$\"$+\"!\"#$\"$0\"!\"#$\"$5\"!\"#$ \"$:\"!\"#$\"$?\"!\"#$\"$D\"!\"#$\"$I\"!\"#$\"$N\"!\"#$\"$S\"!\"#$\"$X \"!\"#$\"$]\"!\"#$\"$b\"!\"#$\"$g\"!\"#$\"$l\"!\"#$\"$q\"!\"#$\"$v\"! \"#$\"$!=!\"#$\"$&=!\"#$\"$!>!\"#$\"$&>!\"#$\"$+#!\"#" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "rest art: with(plots, display):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 7 "n := 7:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 70 "UsualL:=( n-1)*(1/ChisquareP(n-1,alpha/2)-1/ChisquareP(n-1,1-alpha/2));" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 23 "f:=ChisquarePDF(n-1,x):" } }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "eq1 := int(f, x=a..b)=1- alpha:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 37 "eq2:=a^2*subs(x =a,f)=b^2*subs(x=b,f):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 28 "MinLenArray := array(1..39):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "for i from 2 to 40 do" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 97 " eqns := subs(alpha=0.005*i, \{eq1, eq2\});\nSol := fsolve(eqns, \{a,b \}, \{a=0..n-1, b=n-1..infinity\});" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 12 "assign(Sol);" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 47 "MinLenArray [i-1] := [0.005*i, (n-1)*(1/a-1/b)];" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 15 "a:='a'; b:='b';" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 3 "od:" }} }{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 41 "MinLenList := convert(Min LenArray, list);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 46 "A := \+ plot(UsualL, alpha=0.01..0.2, color=red):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 32 "B:=plot(MinLenList, color=blue):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 15 "display(\{A,B\});" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 101 "Improv := [seq([0.005+0.005*i,(value(subs( alpha=0.005+0.005*i,UsualL)-MinLenList[i][2]))], i=1..39)]:" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 150 "PercImprov := [seq([0.005 +0.005*i, 100*(value(subs(alpha=0.005+0.005*i,UsualL))-MinLenList[i][2 ])/value(subs(alpha=0.005+0.005*i,UsualL))], i=1..39)]:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 260 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 268 22 " 10. Order Statistics" }}{PARA 260 "" 0 "" {TEXT -1 30 "restart: with(plots, display):" }}{EXCHG {PARA 260 " " 0 "" {TEXT -1 42 "A study of the p.d.f.s of order statistics" }} {PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 26 "First, take example 10.1-3" }} {PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 12 "f := x -> 1;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 28 "F := x -> int(f(t), t=0..x);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 72 "g := (n, r, y) -> (n!/((r-1)!*(n-r)!)) * F(y)^(r-1)*( 1-F(y))^(n-r)*f(y);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 65 "G: = (n,r,y) -> sum(n!/(k!*(n-k)!)*F(y)^k*(1-F(y))^(n-k), k=r..n);" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 16 "pdf := g(4,3,y);" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "int(pdf, y=1/3..2/3);" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "G(4,3,2/3)-G(4,3,1/3);" }} }{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 50 "Note that this \"hides the expressions for g and G." }}{PARA 260 " " 0 "" {TEXT -1 36 "We get around this by the following." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }} {PARA 260 "" 0 "" {TEXT -1 62 "Can this be done if we assume that the \+ underlying distribution" }}{PARA 260 "" 0 "" {TEXT -1 51 "is more comp licated? Assume we have an exponential." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 18 "assume(theta > 0); " }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 34 "f := x-> ExponentialP DF(theta, x);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 27 "F := x-> int(f(t), t=0..x);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 72 "g \+ := (n, r, y) -> (n!/((r-1)!*(n-r)!)) * F(y)^(r-1)*(1-F(y))^(n-r)*f(y); " }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 65 "G:= (n,r,y) -> sum(n! /(k!*(n-k)!)*F(y)^k*(1-F(y))^(n-k), k=r..n);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 55 "We can now compute probabilities. For example, if n=4, " }}{PARA 260 "" 0 "" {TEXT -1 57 "what is the probability that the third order statistic is" }} {PARA 260 "" 0 "" {TEXT -1 18 "between 1/2 and 1." }}{PARA 260 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 35 "P := si mplify(G(4,3,1)-G(4,3,1/2));" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 " " }}{PARA 260 "" 0 "" {TEXT -1 53 "If we assume a specific theta such \+ as theta=1, we get" }}{PARA 260 "" 0 "" {TEXT -1 17 "a numeric answer. " }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 23 "evalf(subs(theta=1,P));" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 48 "We can now study the ef fect of theta on an order" }}{PARA 260 "" 0 "" {TEXT -1 48 "statistic \+ for specific values of n and r. Let's " }}{PARA 260 "" 0 "" {TEXT -1 14 "take n=4, r=3." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 26 "pdf := simplify(g(4,3,y));" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 7 "N := 5;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to N do" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "gg := simplify(subs(theta=i, g(4,3,y)));" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 38 "P||i := plot(gg, y=0..10, color=blue): " }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 46 "display([seq(P||i, i=1..5)], insequence=true);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 53 "For a fixed theta, say theta=1, we can also consider " }}{PARA 260 "" 0 "" {TEXT -1 53 "the shapes of the order statistics for r=1, 2 , 3, ..." }}{PARA 260 "" 0 "" {TEXT -1 51 "Taking theta =1 and n=4, we look at r=1,2,3, and 4." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to 4 do" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 40 "gg := simplify(subs(theta=1, g(4,i,y))) ;" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 37 "P||i := plot(gg, y=0..4, col or=blue):" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 23 "display(\{P1,P2,P3,P4\});" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 59 "Unfor tunately, the expressions for g and G in both examples" }}{PARA 260 " " 0 "" {TEXT -1 59 "\"hide\" their actual forms. The following will pr oduce more " }}{PARA 260 "" 0 "" {TEXT -1 58 "explicit representations of g and G, if that is desired. " }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 19 "assume( theta > 0);" }} }{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "f := ExponentialPDF(theta , y);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 15 "int(f, y=0..t); " }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 17 "F := subs(t=y,%);" }} }{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 50 "g := (n!/((r-1)!*(n-r)!)) * F^(r-1)*(1-F)^(n-r)*f;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 49 "G := sum(n!/(k!*(n-k)!)*F^k*(1-F)^(n-k), k=r..n);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 55 "For c ontrasrt, we now consider order statistics from a " }}{PARA 260 "" 0 " " {TEXT -1 49 "symmetric, two-parameter distribution: N(mu,var)." }} {PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 18 "assume(sigma > 0);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 31 "f := NormalPDF(mu, sigma^2, y);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 23 "int(f, y=-infinity..t);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 17 "F := subs(t=y,%);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 50 "g := (n!/((r-1)!*(n-r)!)) * F^(r-1)*(1-F)^( n-r)*f;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 59 "G := simplify( sum(n!/(k!*(n-k)!)*F^k*(1-F)^(n-k), k=r..n));" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 44 "As before, we can \+ calculate probabilities in" }}{PARA 260 "" 0 "" {TEXT -1 15 "specific \+ cases." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 " " {MPLTEXT 1 0 51 "GG := simplify(subs(\{n=4, r=3, mu=0, sigma=1\}, G) );" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 41 "evalf(subs(y=1/2, G G)- subs(y=-1/2, GG));" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }} {PARA 260 "" 0 "" {TEXT -1 44 "And we can obtain expressions and graph s for" }}{PARA 260 "" 0 "" {TEXT -1 47 "specific p.d.f.s of order stat istics or through" }}{PARA 260 "" 0 "" {TEXT -1 44 "animation observe \+ the effect of the symmetry" }}{PARA 260 "" 0 "" {TEXT -1 44 "of the un derlying distribution by comparing " }}{PARA 260 "" 0 "" {TEXT -1 19 " g_i with g_(n-i+1)." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 51 "gg := simplify(subs(\{n=4, r=3, m u=0, sigma=1\}, g));" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "p lot(gg, y=-3.5..3.5);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 8 "N := 10;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 t o N do" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 50 "gg := simplify(subs(\{n =N, r=i, mu=0, sigma=1\},g));" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 42 " P||i := plot(gg, y=-3.5..3.5, color=blue):" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 46 "di splay([seq(P||i, i=1..N)], insequence=true);" }}}{EXCHG {PARA 260 "> \+ " 0 "" {MPLTEXT 1 0 6 "N:=20;" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "for i from 1 to N/2 do" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 50 "gg := simplify(subs(\{n=N, r=i, mu=0, sigma=1\},g));" }}{PARA 260 "> \+ " 0 "" {MPLTEXT 1 0 104 "ggg := simplify(subs(\{n=N, r=N-i+1, mu=0, si gma=1\},g));\nP||i := plot(\{gg,ggg\}, y=-3.5..3.5, color=blue):" }} {PARA 260 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 260 "> " 0 " " {MPLTEXT 1 0 48 "display([seq(P||i, i=1..N/2)], insequence=true);" } }}}{SECT 1 {PARA 258 "" 0 "" {TEXT 269 15 " 11. Regression" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 17 "X := [1,2,3,4,5];" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 17 " Y := [2,3,1,5,4];" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "r:= \+ Correlation(X,Y);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 52 "A := BivariateNormalS(100, 225, 400, 625, 0.9, 200):" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 8 "A[1..5];" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 20 "r := Correlation(A);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 18 "y := LinReg(A, x);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 22 "Y := PolyReg(A, 3, x);" }}}{EXCHG {PARA 260 "> " 0 " " {MPLTEXT 1 0 12 "ScatPlot(A);" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 16 "ScatPlotLine(A);" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }} }}{MARK "0 2 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }