{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "Maple Outpu t" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 257 0 "" }{TEXT 258 46 "CHAPTER 3 - Random Variables and Distributions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 259 0 "" }{TEXT 260 21 "Section 3.1, page 102" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 9 "Problem \+ 2" }}{PARA 0 "" 0 "" {TEXT -1 13 "We must have " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Sum(c* x,x=1..5)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&%\"cG\"\" \"%\"xGF)/F*;F)\"\"&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "s olve(%,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$-%$SumG6$%\"xG /F(;F$\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"#:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 45 "So the probability distribution is P(x)=x/15." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 262 9 "Problem 8" }}{PARA 0 "" 0 "" {TEXT -1 79 "This is about t he binomial distribution. With Maple, we can do it the hard way:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "sum(binomial(20,k)*(1/10)^k* (9/10)^(20-k),k=4..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"5px]LMMK `H8\"6++++++++++\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MK`H8!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "The probability is a little more than 13% that \+ more than three of the balls will be red." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 263 0 "" }{TEXT 264 21 "Section 3.2, page 109" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 9 " Problem 4" }}{PARA 0 "" 0 "" {TEXT -1 22 "First, the constant c:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(int(c*x^2,x=1..2)=1,c) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 15 "Now the sketch:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot(3/7*x^2,x=1..2,color=blue,thickness=2,view=[0..3 ,0..2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 221 186 186 {PLOTDATA 2 "6'-%'C URVESG6#7S7$$\"\"\"\"\"!$\"3[&G9dG9dG%!#=7$$\"3hmm;arz@5!#<$\"3g]=%oC$ euWF-7$$\"3OL$e9ui2/\"F1$\"3.5P.g.BUYF-7$$\"3smm\"z_\"4i5F1$\"3A7,JW.X M[F-7$$\"3qmmT&phN3\"F1$\"3`w\"=Lj#)=.&F-7$$\"3UL$e*=)H\\5\"F1$\"3>P)* 4#f*HK_F-7$$\"3sm;z/3uC6F1$\"3w'f9%[!3;U&F-7$$\"3-+]7LRDX6F1$\"3R37-?5 <@cF-7$$\"3em;zR'ok;\"F1$\"3H*3YEB`8$eF-7$$\"3-+]i5`h(=\"F1$\"3!Q\\K,a +Z/'F-7$$\"3YLL$3En$47F1$\"3Q9lY#f`\"oiF-7$$\"3cmmT!RE&G7F1$\"3[-B2%RI $okF-7$$\"3)*****\\K]4]7F1$\"3/a2\\:oW(p'F-7$$\"3))****\\PAvr7F1$\"3hS #)z'3;:$pF-7$$\"3/++]nHi#H\"F1$\"3GS2va\"*)3;(F-7$$\"3bm;z*ev:J\"F1$\" 3?>eWZlTstF-7$$\"3ELL$347TL\"F1$\"3*>4!*QI]zi(F-7$$\"3=LLLjM?`8F1$\"3[ #**F-7$$\"3%)**\\(=>Y2a\"F1$\"3*Hc/>@&Q<5F17$$\"3imm \"zXu9c\"F1$\"36\\$o6#R%\\/\"F17$$\"3'******\\y))Ge\"F1$\"3#)*=05`,Q2 \"F17$$\"3!****\\i_QQg\"F1$\"3!3L2kV8C5\"F17$$\"3#***\\7y%3Ti\"F1$\"3u :Kf1]XI6F17$$\"3#****\\P![hY;F1$\"3ckl#3*H+i6F17$$\"3ELLLQx$om\"F1$\"3 !>z86f?2>\"F17$$\"3')****\\P+V)o\"F1$\"3%*yml5(px]\"F17$$\"3'******\\Qk\\*=F1$\"3-7O*e'G&* Q:F17$$\"3@LL3dg6<>F1$\"3mi9rKJ9v:F17$$\"3_mmmw(Gp$>F1$\"3?)*oPl%oyg\" F17$$\"3-+]7oK0e>F1$\"3&p*\\j968V;F17$$\"3-+](=5s#y>F1$\"3mbs;=-Cx;F17 $$\"\"#F*$\"3>9dG9dG9 " 0 "" {MPLTEXT 1 0 22 "int(3/7*x^2,x=3/2..2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"#P\"#c" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 9 "Problem 8 " }}{PARA 0 "" 0 "" {TEXT -1 23 "This is like problem 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "solve(int(c*exp(-2*x),x=0..infinity )=1,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot(2*exp(-2*x),x=0..6,color=blue,thickness= 2);" }}{PARA 13 "" 1 "" {GLPLOT2D 206 199 199 {PLOTDATA 2 "6'-%'CURVES G6#7Z7$$\"\"!F)$\"\"#F)7$$\"3')*****\\7t&pK!#>$\"3=J7z;9St=!#<7$$\"3s* *****\\i9RlF/$\"3='p(3)*F/$\"3b%)f.(zPPk\"F2 7$$\"3%*******\\#HyI\"!#=$\"3Ib-!*e.pR:F27$$\"3-++voozw=F@$\"3'eY$))eY 3u8F27$$\"33++]([kdW#F@$\"3s![Dgyn&F@7$$\"3e****\\(G[W[(F@$\"3;d [*e.0lZ%F@7$$\"3i****\\()fB:()F@$\"3-iD2(QF(*\\$F@7$$\"39++](Q=\"))**F @$\"34/6i=`98FF@7$$\"3(****\\P'=pD6F2$\"3uHvFK'p]5#F@7$$\"33+++lN?c7F2 $\"3()[ChDqX@;F@7$$\"3-++]U$e6P\"F2$\"3u/kh<'=%)G\"F@7$$\"36+++&>q0]\" F2$\"3+h#))3\"[1Y**F/7$$\"3'******\\U80j\"F2$\"3sIudC)*zpwF/7$$\"35+++ 0ytb@F2$\"3wW*\\Ja7g)GF/7$$\"3'**** \\(3wY_AF2$\"3>x@&yDh3@#F/7$$\"3#)******HOTqBF2$\"3G9$y%3SGY)*\\#F2$\"3s]/GAq2[8F/7$$\"3:++DEP/BEF2$\"3GJvBczh`5F/7$$\"3=++ ](o:;v#F2$\"3IiJ2APFhw7$$\"3u*****\\KCnu%F2$\"3]2kG0k(o]\"F hw7$$\"3s***\\(=n#f([F2$\"3;z1ilVtj6Fhw7$$\"3P+++!)RO+]F2$\"3iN&)>\"\\ yL2*!#A7$$\"30++]_!>w7&F2$\"3d?F_`dbMqFfz7$$\"3O++v)Q?QD&F2$\"3?GkiY-P laFfz7$$\"3G+++5jyp`F2$\"38/8tO&RSL%Ffz7$$\"3<++]Ujp-bF2$\"3M&[&*)Q_PA LFfz7$$\"3++++gEd@cF2$\"3+sRBgHN>EFfz7$$\"39++v3'>$[dF2$\"3;U0h#=RG.#F fz7$$\"37++D6EjpeF2$\"3E+B%RB%*[f\"Ffz7$$\"\"'F)$\"3'>kl1ZU)G7Ffz-%'CO LOURG6&%$RGBGF(F($\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!Fj]l-%*THICKNE SSG6#F+-%%VIEWG6$;F(F[]l%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "int(2*exp(-2*x),x=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#!\"%!\"\"-F%6#!\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+Vk>q6!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "So the probabili ty that we will find X between 1 and 2 is about 11.7 %" }}{PAGEBK } {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 267 0 "" }{TEXT 268 21 "Section 3.3, page 116" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 9 "Problem 8 " }}{PARA 0 "" 0 "" {TEXT -1 121 "We start with the cumulative distrib ution function of Z. First, note that the pdf on the disk is uniform w ith a value of " }{XPPEDIT 18 0 "1/Pi;" "6#*&\"\"\"F$%#PiG!\"\"" } {TEXT -1 38 ", since the area of the whole disk is " }{XPPEDIT 18 0 "P i;" "6#%#PiG" }{TEXT -1 71 ". So the probability that a point is withi n r units from the origin is " }{XPPEDIT 18 0 "Pi*r^2/Pi;" "6#*(%#PiG \"\"\"*$%\"rG\"\"#F%F$!\"\"" }{TEXT -1 12 ", or simply " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT -1 146 ". Since the pdf of the distri bution of Z is simply the derivative of its cumulative distribution, w e have that the df of Z is 2r. 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