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Random Variable: A" }{TEXT 274 16 " random variable" } {TEXT -1 38 " is a continuous variable that ranges " }}{PARA 0 "" 0 " " {TEXT -1 101 " over an interval. For example, the height of an \+ adult individual - it lies between 36 and 85. " }}{PARA 0 "" 0 "" {TEXT -1 31 "2. For every random variable " }{TEXT 277 2 "X " } {TEXT -1 44 " there is an associated function called the " }{TEXT 275 28 "probability density function" }{TEXT -1 13 ", written as " }{TEXT 259 3 "pdf" }{TEXT -1 2 " (" }{TEXT 276 1 "X" }{TEXT -1 47 "). The co nnection between the random variable " }{TEXT 278 1 "X" }{TEXT -1 16 " and its pdf, f(" }{TEXT 279 1 "X" }{TEXT -1 15 "), is given by:" }} {PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "int(f(x),x = a \+ .. b);" "6#-%$intG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 22 " = Pro bability that " }{TEXT 280 1 "X" }{TEXT -1 25 " lies in the interval (" }{TEXT 260 4 "a, b" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 56 " Geometrically, this probability is the area under " }{TEXT 281 2 "y " }{TEXT -1 4 "= f(" }{TEXT 261 1 "x" }{TEXT -1 7 ") from " } {TEXT 262 1 "a" }{TEXT -1 4 " to " }{TEXT 263 1 "b" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 38 "with the following properties: a) f(" }{TEXT 264 1 "x" }{TEXT -1 16 ") >= 0 for all " }{TEXT 282 1 "x" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 53 " \+ b) " }{XPPEDIT 18 0 "int(f(x),x = -infinity .. infinity)" "6#-%$intG6$-%\"fG6#%\"xG/F);,$%)infinityG!\"\"F-" } {TEXT -1 36 " = 1 --i.e., the probability that " }{TEXT 283 2 " X" } {TEXT -1 12 " must have " }}{PARA 0 "" 0 "" {TEXT -1 122 " \+ some value on the real num ber line is 1 or is a certain occurrence." }}{PARA 0 "" 0 "" {TEXT -1 57 " " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 265 18 " Solved Problem 1:" }}{PARA 0 "" 0 "" {TEXT -1 25 " 1. \+ Plot the graph of " }{XPPEDIT 18 0 "1*e^(-x^2/2)/sqrt(2*Pi);" "6#*( \"\"\"F$)%\"eG,$*&%\"xG\"\"#F*!\"\"F+F$-%%sqrtG6#*&F*F$%#PiGF$F+" } {TEXT -1 16 " and evaluate " }{XPPEDIT 18 0 "int(1*e^(-x^2/2)/sqrt(2 *Pi),x = -3 .. 3);" "6#-%$intG6$*(\"\"\"F')%\"eG,$*&%\"xG\"\"#F-!\"\"F .F'-%%sqrtG6#*&F-F'%#PiGF'F./F,;,$\"\"$F.F7" }{TEXT -1 8 " and " } {XPPEDIT 18 0 "int(1*e^(-x^2/2)/sqrt(2*Pi),x = -infinity .. infinity); " "6#-%$intG6$*(\"\"\"F')%\"eG,$*&%\"xG\"\"#F-!\"\"F.F'-%%sqrtG6#*&F-F '%#PiGF'F./F,;,$%)infinityGF.F7" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 25 "2. Plot the graph of " }{XPPEDIT 18 0 "1*e^(-(x-2)^2/2)/sqrt(2*Pi);" "6#*(\"\"\"F$)%\"eG,$*&,&%\"xGF$\"\"# !\"\"F+F+F,F,F$-%%sqrtG6#*&F+F$%#PiGF$F," }{TEXT -1 16 " and evaluate " }{XPPEDIT 18 0 "int(1*e^(-(x-2)^2/2)/sqrt(2*Pi),x = -1.5 .. 5.5); " "6#-%$intG6$*(\"\"\"F')%\"eG,$*&,&%\"xGF'\"\"#!\"\"F.F.F/F/F'-%%sqrt G6#*&F.F'%#PiGF'F//F-;,$-%&FloatG6$\"#:F/F/-F96$\"#bF/" }}{PARA 0 "" 0 "" {TEXT -1 25 "3. Plot the graph of " }{XPPEDIT 18 0 "1*e^(-(x- 2)^2/(2(.5)^2))/(sqrt(2*Pi)*.5);" "6#*(\"\"\"F$)%\"eG,$*&,&%\"xGF$\"\" #!\"\"F+*$-F+6#-%&FloatG6$\"\"&F,F+F,F,F$*&-%%sqrtG6#*&F+F$%#PiGF$F$-F 16$F3F,F$F," }{TEXT -1 16 " and evaluate " }{XPPEDIT 18 0 "int(1*e^( -(x-2)^2/(2*.5^2))/(sqrt(2*Pi)*.5),x = -1 .. 5);" "6#-%$intG6$*(\"\"\" F')%\"eG,$*&,&%\"xGF'\"\"#!\"\"F.*&F.F'*$-%&FloatG6$\"\"&F/F.F'F/F/F'* &-%%sqrtG6#*&F.F'%#PiGF'F'-F36$F5F/F'F//F-;,$F'F/F5" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }} {PARA 0 "" 0 "" {TEXT -1 113 "First note that each of the above functi ons is positive, since an exponential function is never zero or negati ve." }}{PARA 0 "" 0 "" {TEXT -1 3 "1. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot( 1*exp(-(x-2)^2/ (2(.5)^2))/(sqrt(2*Pi)*.5),x=-1..5);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 249 171 171 {PLOTDATA 2 "6%-%'CURVESG6$7W7$$!\"\"\"\"!$\"3ur0&)*RT'4%) !#>7$$!3/+++]2<#p)!#=$\"321>3vy')=5F17$$!3[++]7bBavF1$\"3c>hyynm&>\"F1 7$$!3++++D$3XF'F1$\"3g6y3cEP?9F17$$!3c*****\\F)H')\\F1$\"3?j%3p+B`n\"F 17$$!3J++]i3@/PF1$\"357:T6iGe>F17$$!3V++]7I\"GK2d#F17$$!33')****\\7;)=\"!#?$\"3$=PUcum<$HF17$$ \"3m****\\P'=pD\"F1$\"38!3PW!)f_J$F17$$\"3y+++]c.iDF1$\"3P)Gl%[xmIPF17 $$\"3;+++DMe6PF1$\"33b3SbvO5TF17$$\"32,++]>q0]F1$\"3]gn./M:[XF17$$\"3h ******\\U80jF1$\"3gvl6X!=C*\\F17$$\"3'4+++0ytb(F1$\"3^)>,zy2\"=aF17$$ \"3w****\\(QNXp)F1$\"3%RjgDiKlz&F17$$\"3.+++XDn/5!#<$\"3FZtw>;VGiF17$$ \"3.+++!y?#>6Fhp$\"3o/0,,Q@slF17$$\"3'****\\(3wY_7Fhp$\"3V>t\">G[&QpF1 7$$\"3#)******HOTq8Fhp$\"37B3tFl4EsF17$$\"37++v3\">)*\\\"Fhp$\"3g%**)> sF4&\\(F17$$\"3:++DEP/B;Fhp$\"33oBl:/Q+xF17$$\"3=++](o:;v\"Fhp$\"3#)H5 !y`En&yF17$$\"3>+]i&G]1\"=Fhp$\"3UFK*G:[w!zF17$$\"3=++v$)[op=Fhp$\"3Gx >9lJ/XzF17$$\"32+]7t;OL>Fhp$\"3gvmtuE**pzF17$$\"3%*****\\i%Qq*>Fhp$\"3 'e$Q-8\"G)yzF17$$\"3%***\\i]2=j?Fhp$\"3SG%pj2()3(zF17$$\"3&****\\(QIKH @Fhp$\"3Y`t*\\![bXzF17$$\"3#******\\4+p=#Fhp$\"3*\\M<5Uq%4zF17$$\"3#** **\\7:xWC#Fhp$\"3YM#*z^)40'yF17$$\"37++]Zn%)oBFhp$\"30sV'ySK?r(F17$$\" 3y******4FL(\\#Fhp$\"3t'o'RR#=/](F17$$\"3#)****\\d6.BEFhp$\"3xHd^Za%4C (F17$$\"3(****\\(o3lWFFhp$\"3_>xa93,YpF17$$\"3!*****\\A))ozGFhp$\"3s5% )pz&o`d'F17$$\"3e******Hk-,IFhp$\"36L8>28u5iF17$$\"36+++D-eIJFhp$\"3[U k#3*>U'z&F17$$\"3u***\\(=_(zC$Fhp$\"38U>zRye0aF17$$\"3M+++b*=jP$Fhp$\" 3GE#>^%z0p\\F17$$\"3g***\\(3/3(\\$Fhp$\"3ea:z!>lhb%F17$$\"33++vB4JBOFh p$\"3?s:s`A*)GTF17$$\"3u*****\\KCnu$Fhp$\"3Vb*>%)=[6s$F17$$\"3s***\\(= n#f(QFhp$\"3jd2dfEB5LF17$$\"3P+++!)RO+SFhp$\"3n&R?s)\\=MHF17$$\"30++]_ !>w7%Fhp$\"3qQ2*[ZyId#F17$$\"3O++v)Q?QD%Fhp$\"3hx[Gdm(3C#F17$$\"3G+++5 jypVFhp$\"3sf\"fA()f(f>F17$$\"3<++]Ujp-XFhp$\"3G#GZ\"fk#om\"F17$$\"3++ ++gEd@YFhp$\"3)R_4BIr8V\"F17$$\"39++v3'>$[ZFhp$\"3RV@x57T27F17$$\"37++ D6Ejp[Fhp$\"3]Ig`K1E=5F17$$\"\"&F*F+-%'COLOURG6&%$RGBG$\"#5F)$F*F*Fc\\ l-%+AXESLABELSG6$Q\"x6\"Q!Fh\\l-%%VIEWG6$;F(F[\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " } {TEXT 257 17 "Solved Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 46 "For An chorage, Alaska its rainfall in inches, " }{TEXT 287 1 "X" }{TEXT -1 165 ", can be considered a normally distributed random variable with d ensity function p(x) = (1/sqrt(2Pi))exp(-(x-15)^2/2). The mean is 1 5 and standard deviation is 1." }}{PARA 0 "" 0 "" {TEXT -1 103 " \+ (a) Compute the probability that the rainfall in a given year is between 13 and 16 inches. " }}{PARA 0 "" 0 "" {TEXT -1 78 " \+ (b) Plot the normal density and cumulative distribution functions." }}{PARA 0 "" 0 "" {TEXT -1 51 " (c) Find the smallest posit ive number " }{TEXT 288 1 "k" }{TEXT -1 53 " so that you can be 98% \+ sure that the rainfall next " }}{PARA 0 "" 0 "" {TEXT -1 56 " \+ year in Anchorage will be between 15 - " }{TEXT 289 1 "k" } {TEXT -1 12 " and 15 + " }{TEXT 290 1 "k" }{TEXT -1 8 " inches." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "So lution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 268 56 " (a) Probability of the rainfall between \+ 13 to 16 inches" }}{PARA 0 "" 0 "" {TEXT -1 31 " First we define the f unction " }{TEXT 291 1 "p" }{TEXT -1 32 " in Maple. Then we integrat e p(" }{TEXT 292 1 "x" }{TEXT -1 23 ") between 13 and 16. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "p := t -> (1/sqrt(2*Pi))*exp (-(t-15)^2/2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"tG6\" 6$%)operatorG%&arrowGF(*&-%%sqrtG6#,$*&\"\"#\"\"\"%#PiGF3F3!\"\"-%$exp G6#,$*&#F3F2F3*$),&9$F3\"#:F5F2F3F3F5F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "int (p(t), t = 13..16);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%$erfG6#,$*&F'!\"\"F'F%F&F&F&*&F%F&- F)6#*$F'F%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(%); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Sh%f=)!#5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 266 72 " (b) Graphs of probability density and cumulative distribution fun ctions" }}{PARA 0 "" 0 "" {TEXT -1 48 "The cumulative distribution fu nction is P(x) = " }{XPPEDIT 18 0 "int(p(t),t = -infinity .. x);" "6#- %$intG6$-%\"pG6#%\"tG/F);,$%)infinityG!\"\"%\"xG" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "P := x -> int(p(t), t = -infinity..x);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGf*6#%\"xG6\"6$%)operatorG%&arrow GF(-%$intG6$-%\"pG6#%\"tG/F2;,$%)infinityG!\"\"9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(p(x), x = 10..20);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 237 148 148 {PLOTDATA 2 "6%-%'CURVESG6$7eo7$$\"#5 \"\"!$\"3o(HMZ^>n[\"!#B7$$\"3vmm;arz@5!#;$\"3C0ca)z)[\"3MFF97$$\"3um mT&phN3\"F1$\"3o(*>(4(z1ToF97$$\"3CL$e*=)H\\5\"F1$\"3%R/4&RU/G;!#@7$$ \"3jm;z/3uC6F1$\"3WP&GjWR=\\$FI7$$\"3)***\\7LRDX6F1$\"3?3,!H#e(GQ(FI7$ $\"3km;zR'ok;\"F1$\"3$zpS`4H@`\"!#?7$$\"3%***\\i5`h(=\"F1$\"3g)e(3Ti;L IFY7$$\"3PLL$3En$47F1$\"3%p0F6$e=WeFY7$$\"3ummT!RE&G7F1$\"3Wiy55lL,5!# >7$$\"3)*****\\K]4]7F1$\"3S!fTBg)*pv\"Fco7$$\"3)*****\\PAvr7F1$\"3-E@0 I[u[HFco7$$\"3-++]nHi#H\"F1$\"3w)H;:nHek%Fco7$$\"3tm;z*ev:J\"F1$\"3%o \\O5\"49gnFco7$$\"3SLL$347TL\"F1$\"3o_M)>TOx+\"!#=7$$\"3ZLL3xxlV8F1$\" 3;0r!)R[Fv6F]q7$$\"3QLLLjM?`8F1$\"3;a;1,TBe8F]q7$$\"3em\"HdO2VO\"F1$\" 3u,p08$o))e\"F]q7$$\"3)***\\7o7Tv8F1$\"3U>5#HA*)e$=F]q7$$\"3om\"HK5S_Q \"F1$\"3E#[m\"y30l?F]q7$$\"3OLLLQ*o]R\"F1$\"3G$*e?qa[+BF]q7$$\"3sm\"H# GF&eS\"F1$\"3i3)[4:i6c#F]q7$$\"32+]7=lj;9F1$\"3jEKlX/T=GF]q7$$\"3,++DO _!pU\"F1$\"3w$*H#\\?fT0$F]q7$$\"3&***\\PaRvRF]q7$$\"3?$3-8,+U\\\"F1 $\"3=D()3h#=F)RF]q7$$\"3)[Pf3@`o\\\"F1$\"3OQGXt![u)RF]q7$$\"3emmT5k]* \\\"F1$\"3kO9-2UP*)RF]q7$$\"3fT&)3RBE-:F1$\"37PR!e+-%))RF]q7$$\"3h;/wn #=]]\"F1$\"3wq?C\"o-W)RF]q7$$\"3j\"HKk>ux]\"F1$\"3;'G'[Q`QxRF]q7$$\"3k mT5D,`5:F1$\"3A[P**QfOnRF]q7$$\"3o;zW#)>/;:F1$\"3mI^c3(=%QRF]q7$$\"3sm ;zRQb@:F1$\"3))y&\\>(G#y*QF]q7$$\"3YLL$e,]6`\"F1$\"3<;K(HW\"\\+QF]q7$$ \"3-+](=>Y2a\"F1$\"3Og*4PcB;n$F]q7$$\"3_Le*[K56b\"F1$\"3kmf+oi%4]$F]q7 $$\"3%om;zXu9c\"F1$\"3oU*p`\\PDI$F]q7$$\"3[L$e9i\"=s:F1$\"31'GiCZ%[uIF ]q7$$\"35+++&y))Ge\"F1$\"3e%)[N%y`&HGF]q7$$\"31+]ibOO$f\"F1$\"3#fVFF'y /!e#F]q7$$\"3.++DE&QQg\"F1$\"3s$o?wJroK#F]q7$$\"38+v=-N(Rh\"F1$\"3o#Gb #orq$3#F]q7$$\"3C+]7y%3Ti\"F1$\"3o^ky]d)o%=F]q7$$\"3.+v$4kh`j\"F1$\"3A QJ!3w6gf\"F]q7$$\"3#)***\\P![hY;F1$\"3-3BZa,'=O\"F]q7$$\"3Qm;/risc;F1$ \"3Sf1+s1Bo6F]q7$$\"3ELLLQx$om\"F1$\"3rRdb$*oP>**Fco7$$\"3C++]P+V)o\"F 1$\"3GCCnZTUfnFco7$$\"3im;zpe*zq\"F1$\"3=M+)\\i,le%Fco7$$\"37++]#\\'QH dc/W!H(GFco7$$\"3AL$e9S8&\\(Q?uFI7$$\"3kL$e9tOc(=F1$\"3D1w8EMpUMFI7$$\"35+++&Qk \\*=F1$\"3OzX7zA'[j\"FI7$$\"3mLL3dg6<>F1$\"3_J'oe%3`]mF97$$\"3hmmmw(Gp $>F1$\"3aV#e8V$z`GF97$$\"3')**\\7oK0e>F1$\"3YXc0Wi()36F97$$\"3I+](=5s# y>F1$\"3G*pm/d7KI%F-7$$\"#?F*F+-%'COLOURG6&%$RGBG$F)!\"\"$F*F*Fjbl-%+A XESLABELSG6$Q\"x6\"Q!F_cl-%%VIEWG6$;F(Fbbl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(P(x), x = 0..40);\n" }}{PARA 13 "" 1 " " {GLPLOT2D 230 132 132 {PLOTDATA 2 "6%-%'CURVESG6$7_o7$$\"\"!F)F(7$$ \"3Hmmmm;')=()!#=F(7$$\"3RLLLe'40j\"!#(>%F1F(7$$\"3%omm;>K'*)\\F1F(7$$\"3g*****\\Kd, \"eF1F(7$$\"3Onmm\"fX(emF1F(7$$\"3O+++DCh/vF1$\"3rS)f#)\\8HI$!#J7$$\"3 uMLLL/pu$)F1$\"3<'*[6,&3Et\"!#G7$$\"3mnmm;c0T\"*F1$\"3P*R2rI$4HB!#E7$$ \"35+++I,Q+5!#;$\"3$)\\'3;uqN#H!#C7$$\"31+++]*3q3\"FY$\"3R'[FbTAX\"=!# A7$$\"3/+++q=\\q6FY$\"3DaLB_/m>\\!#@7$$\"3[LLe9rR37FY$\"3W&GqTc)es 7$$\"33++Dh`P\"H\"FY$\"3e62f9b\"y%=Fcp7$$\"3um;Hi=\"RJ\"FY$\"33P![0^Y! QJFcp7$$\"3TLLLj$[kL\"FY$\"3/Y!4%GB/(4&Fcp7$$\"3aLLL36ju8FY$\"3Cy4#**[ v(\\5F-7$$\"3ZLLL`Q\"GT\"FY$\"3\"4Hjjy>k\">F-7$$\"32+]7e;-N9FY$\"3q\"3 m[lg\"zDF-7$$\"3mmm\"HYHsX\"FY$\"3Spmt&>GVM$F-7$$\"3WL$3xEP%z9FY$\"3eT 'H\\;5a=%F-7$$\"3.++]s]k,:FY$\"3#e?Eu$fil]F-7$$\"3EL$3Fu-8_\"FY$\"3R%z \\.:sM%eF-7$$\"3mmm\"HTg4a\"FY$\"3)))H^G;=&*e'F-7$$\"31+]7$3=1c\"FY$\" 3ELs1*fE!ysF-7$$\"3GLLL`dF!e\"FY$\"3g\"3>/aD%*)yF-7$$\"3;+]7LL%=g\"FY$ \"3]rsnb'RwX)F-7$$\"3'om;H\"4TB;FY$\"31tD4*y)=9*)F-7$$\"3dL$3F\\y\\k\" FY$\"33advzuSk#*F-7$$\"3G++]sgam;FY$\"3DAL\"\\j&*3_*F-7$$\"3;++v3N3(o \"FY$\"3iyZ/(GgJp*F-7$$\"3/+++X4i2FY$\"3gcEM)e>)****F-7$$\"3kmmmTc-)*>FY$\"2e$f*\\#o******F17$$\" 3)omm\"f`@'3#FY$\"2_*[:x********F17$$\"31++]nZ)H;#FY$\"2K)>$)********* *F17$$\"3+nmmJy*eC#FY$\"2[c*************F17$$\"3/+++S^bJBFY$\"\"\"F)7$ $\"37+++0TN:CFYFcy7$$\"3A++]7RV'\\#FYFcy7$$\"3++++:#fke#FYFcy7$$\"31LL L`4NnEFYFcy7$$\"3?+++],s`FFYFcy7$$\"3\\mm;zM)>$GFYFcy7$$\"3Z+++qfa " 0 "" {MPLTEXT 1 0 40 "prob := int (p(t), t = 15 - k..15 + k);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%probG-%$erfG6#,$*(\"\"#! \"\"F*#\"\"\"F*%\"kGF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 86 "We want this probability to be 0.98, so we ask Maple to solve the equation prob = \+ .98." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolve(prob = 0.98, \+ k);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+uyMEB!\"*" }}}}}{PARA 0 " " 0 "" {TEXT -1 80 "__________________________________________________ ______________________________" }}{PARA 257 "" 0 "" {TEXT -1 10 "ASSIG NMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 269 10 "Problem 1:" }}{PARA 0 "" 0 "" {TEXT -1 96 " Suppose that the average rainfall in May is 5 inches with a stand ard deviation of 0.6 inches. " }}{PARA 0 "" 0 "" {TEXT -1 90 "What is the probability that the rainfall next May will differ from the avera ge by 1 inch?" }}{PARA 0 "" 0 "" {TEXT -1 49 "Assume that May rainfall is normally distributed." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 270 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 29 " > 0 be a constant, and let " }{TEXT 271 1 "f" }{TEXT -1 19 " \+ be the function " }{TEXT 272 1 "f" }{TEXT -1 1 "(" }{TEXT 295 1 "x" }{TEXT -1 1 ")" }{TEXT 296 3 " = " }{XPPEDIT 18 0 "lambda;" "6#%'lambd aG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "e^(-lambda*x);" "6#)%\"eG,$*&%'lam bdaG\"\"\"%\"xGF(!\"\"" }{TEXT -1 33 " for x >= 0, 0 otherwise ." }}{PARA 0 "" 0 "" {TEXT -1 15 "(a) Show that " }{TEXT 273 1 "f" } {TEXT -1 58 " is a probability density function. (It is known as the " }{TEXT 294 28 "exponential density function" }{TEXT -1 2 ".)" }} {PARA 0 "" 0 "" {TEXT -1 13 "(b) Evaluate " }{XPPEDIT 18 0 "int(x*f(x) ,x = -infinity .. infinity);" "6#-%$intG6$*&%\"xG\"\"\"-%\"fG6#F'F(/F' ;,$%)infinityG!\"\"F/" }{TEXT -1 44 " (The value of this integ ral is the " }{TEXT 298 4 "mean" }{TEXT -1 5 " of " }{TEXT 297 1 "X" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 102 "(c) This model is used to model the lifespan of electrical devices (in months). Suppose tha t 10% of a" }}{PARA 0 "" 0 "" {TEXT -1 67 " certain type of devi ce fail after 6 months. What value of " }{XPPEDIT 18 0 "lambda;" "6# %'lambdaG" }{TEXT -1 29 " corresponds to the observed" }}{PARA 0 "" 0 "" {TEXT -1 16 " lifespan? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 98 "(d) What is the probability that a rando mly selected device with conditions in (c) will still be " }}{PARA 0 " " 0 "" {TEXT -1 37 " functioning after 18 months? " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "(e) What is the me an lifetime of the device? " }}}{PARA 0 "" 0 "" {TEXT -1 79 "____ ______________________________________________________________________ _____" }}{PARA 0 "" 0 "" {TEXT -1 99 "MSIP Grant # P120A80089-98: \"T hree Urban Calculus Reform Programs: Adopting the Best\" 1998-2001 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 11 " " }}} {MARK "24 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }