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Gosset, whose factory prohibited him from publishing papers, so he p ublished his writing under the name \"student\". Thus, the " }{TEXT 263 1 "t" }{TEXT -1 43 " distribution is also known as the Student-" } {TEXT 264 1 "t" }{TEXT -1 27 " distribution or Student's " }{TEXT 265 1 "t" }{TEXT -1 14 " distribution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 266 1 "Y" }{TEXT -1 5 " and " }{TEXT 267 1 "Z" }{TEXT -1 35 " are independent random variables, " } {TEXT 268 2 "Y " }{TEXT -1 6 "has a " }{XPPEDIT 18 0 "chi^2;" "6#*$%$c hiG\"\"#" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "nu;" "6#%#nuG" }{TEXT -1 57 ") distribution, and Z ~ N(0,1), then the distribution of " }{XPPEDIT 269 0 "T = Z/sqrt(Y/nu);" "6#/%\"TG*&%\"ZG\"\"\"-%%sqrtG6#*&%\"YGF'%#n uG!\"\"F." }{TEXT -1 34 " has probability density function " }}{PARA 0 "" 0 "" {XPPEDIT 270 0 "f(t) = Gamma((nu+1)/2)*(1+t^2/nu)^(-(nu+1)/2 )/(sqrt(Pi*nu)*Gamma(nu/2));" "6#/-%\"fG6#%\"tG*(-%&GammaG6#*&,&%#nuG \"\"\"\"\"\"F/F/\"\"#!\"\"F/),&\"\"\"F/*&F'\"\"#F.F2F/,$*&,&F.F/\"\"\" F/F/\"\"#F2F2F/*&-%%sqrtG6#*&%#PiGF/F.F/F/-F*6#*&F.F/\"\"#F2F/F2" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 214 "It was o riginally developed to be used for the distribution of sample means fr om a normal population when the variance of the normal distribution is not known, but estimated with the sample variance. That is, if " } {TEXT 274 2 "X " }{TEXT -1 4 "and " }{XPPEDIT 18 0 "S^2;" "6#*$%\"SG\" \"#" }{TEXT -1 69 " are the mean and variance, respectively, of a rand om sample of size " }{TEXT 271 1 "n" }{TEXT -1 36 " from a normal popu lation with mean " }{XPPEDIT 272 0 "mu;" "6#%#muG" }{TEXT -1 14 " and \+ variance " }{XPPEDIT 18 0 "sigma^2;" "6#*$%&sigmaG\"\"#" }{TEXT -1 7 " , then " }{XPPEDIT 273 0 "T = (X-mu)/(S/sqrt(n));" "6#/%\"TG*&,&%\"XG \"\"\"%#muG!\"\"F(*&%\"SGF(-%%sqrtG6#%\"nGF*F*" }{TEXT -1 9 " has the \+ " }{TEXT 275 2 "t " }{TEXT -1 18 "distribution with " }{XPPEDIT 18 0 " nu;" "6#%#nuG" }{TEXT -1 3 " = " }{TEXT 276 1 "n" }{TEXT -1 39 " -1 de grees of freedom. 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45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(F(-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ ++++D!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The value of this area can also be found by using the int egration command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "evalf(int(f(x),x=-infinity..-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++D!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a:=plot(f(x), x=-6..6, color=black):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "b:=plot(f(x), x=.6..1.4, color=yellow, filled=tr ue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "display([a, b]);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Find the value of the yellow shaded area, the area under the " } {TEXT 287 1 "t" }{TEXT -1 37 "(1) density curve between .6 and 1.4." } }{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7ao 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{XPPMATH 20 "6#> %&area1G$\"+mXoD!)!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ar ea2:=F(0.6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&area2G$\"+'p3-s'!#5 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "area_between:=area1-are a2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-area_betweenG$\"+qeZ08!#5" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 " Using the integration command, we simply integrate the " }{TEXT 288 2 "t " }{TEXT -1 20 "PDF from 0.6 to 1.4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalf(int(f(x),x=0.6.. 1.4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+qeZ08!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "a:=plot(f(x),x=-6..6,labels=[\" \", \" \"]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "b:=plot(f(x),x= 1..6,color=yellow,filled=true,labels=[\" \",\" \"]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "c:=textplot([1,-.01,\"x\"],color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 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1$\"1KEE!=(pj5F97$F\\emF*7&F`emF[em7$$\"1,++D>#[Z&F1$\"1**3**y(yw-\"F9 7$FcemF*7&FgemFbem7$$\"1nmT&G!e&e&F1$\"1$pj17(z&))*F-7$FjemF*7&F^fmFie m7$$\"1MLL$)Qk%o&F1$\"1z`K!4,Xb*F-7$FafmF*7&FefmF`fm7$$\"1+]iSjE!z&F1$ \"1'H[*4$4\">#*F-7$FhfmF*7&F\\gmFgfm7$$\"1,]P40O\"*eF1$\"1()[g.O@9*)F- 7$F_gmF*7&FcgmF^gmFj`l7$F[alF*7\"-%&STYLEG6#%,PATCHNOGRIDG-F^al6&F`al$ \"*++++\"!\")F]hmF*-%%TEXTG6%7$Fial$Fcal!\"#%\"xG-F^al6&F`alF*F*F]hm-F ahm6%7$$\"\"#F*$F]imFehm%$.25G-F^al6&F`alF*F*F*-%+AXESLABELSG6$Q\"~6\" Feim-%%VIEWG6$;F(F[al%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 18 "Find the value of " }{TEXT 286 3 "x, " } {TEXT -1 39 "which has .75 area to the left of it. " }{TEXT 289 1 "x " }{TEXT -1 39 " is known as the third quartile of the " }{TEXT 290 1 "t" }{TEXT -1 17 "(1) distribution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fsolve(F(x)=.75,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++5!\"*" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Properties of the Distribution" }}{EXCHG {PARA 259 " " 0 "" {TEXT -1 0 "" }{TEXT 260 32 "Calculation of Mean and Variance" }}{PARA 0 "" 0 "" {TEXT -1 29 "To calculate the mean of the " }{TEXT 291 1 "t" }{TEXT -1 40 " distribution, we can simply integrate " } {TEXT 261 1 "x" }{TEXT -1 2 "*(" }{TEXT 292 1 "t" }{TEXT -1 54 "PDF), \+ per the definition of mathematical expectation." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=0):" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "When nu=1 the integral of x*f(x) diverges so the expected value does \+ not exist. We consider the case where nu>1 below." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assume(nu > 1) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f(x):=TPDF(nu,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#%\"xG*&-%&GAMMAG6#,&%$nu|irG# \"\"\"\"\"#F.F/\"\"\"*(-F*6#,$F-F.\"\"\"-%%sqrtG6#*&F-F/%#PiGF/F1),&F/ F/*&*$)F'F0F1F1F-!\"\"F/F,\"\"\"FA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "EX:=int(x*f(x),x=-infinity..infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#EXG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Calculating Var(" }{TEXT 258 1 "X" }{TEXT -1 23 "), the variance o f the " }{TEXT 297 1 "t" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "nu;" "6#%#nuG " }{TEXT -1 50 ") distribution. We will employ the formula: Var(" } {TEXT 259 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 85 "The integral of x^2*f(x) div erges if nu <=2. We consider the case where nu>2 below. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assu me(nu > 2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "E_X_SQ:=int((x^2)*f( x),x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'E_X_S QG,$*&%$nu|irG\"\"\",&F'#\"\"\"\"\"#!\"\"F+!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "VarX:=simplify(E_X_SQ-EX^2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%VarXG*&%$nu|irG\"\"\",&F&\"\"\"!\"#F)!\"\"" } }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 2 "Sp" }{TEXT 18 0 "" }{TEXT 257 16 "ecial Properties " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 293 0 "nu;" "6#%#nuG" }{TEXT -1 16 " increases, the \+ " }{TEXT 294 2 "t " }{TEXT -1 144 "distribution approaches the N(0,1) \+ distribution. The next few plots and animations will try to give you \+ a visual representation of this result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "nu:=1:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "f:=x->TPDF(nu, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%%TPDFG6$%#nuG 9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "n:=x->NormalPDF (0,1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nGR6#%\"xG6\"6$%)opera torG%&arrowGF(-%*NormalPDFG6%\"\"!\"\"\"9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([f(x),n(x)],x=-5..5,title=\"t(1) with N( 0,1)\");" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The red plot i s the " }{TEXT 295 1 "t" }{TEXT -1 64 "(1) distribution, and the green plot is the N(0,1) distribution." }}{PARA 13 "" 1 "" {GLPLOT2D 384 384 384 {PLOTDATA 2 "6'-%'CURVESG6$7ao7$$!\"&\"\"!$\"1!e9IzoUA\"!#<7$$ 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