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[r7$FcxFb[r7$FfxFb[r7$FixFb[r7$F\\yFb[r7$F_yFb[r7$FbyFb[r7$FeyFb[r7$F_
jqFb[r7$FhyFb[r7$$\"1](=#HA6^HFfsFb[r7$FejqFb[r7$$\"1v$4Y6cb(HFfsFb[r7
$$\"1]iSwSq$)HFfsFb[r7$$\"1)o/t0yx)HFfsFb[r7$$\"1DJ?Q?&=*HFfsFb[r7$$\"
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FazFgz-%&TITLEG6#Q.Changing~betaFez" 1 2 0 1 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Can you \+
anticipate what happens if " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 294 " \+
both increase at the same rate?  In the following animation, they both
 increase at a step size of 0.1  After you examine this animation, fee
l free to change the code so that they increase at different rates.  I
f you'd like, you can change it so one is decreasing while the other i
s increasing." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "alpha:=0; \+
ainitial:=0; alpha_step:=0.1; beta:=1; binitial:=1; beta_step:=0.1;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)ainitialG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
+alpha_stepG$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)binitialG\"\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%*beta_stepG$\"\"\"!\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 51 "f(x):=piecewise(x>alpha and x<beta,1/(bet
a-alpha));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#%\"xG-%*PIECEWI
SEG6$7$\"\"\"32,$F'!\"\"\"\"!2,&F'F,F0F,F17$F1%*otherwiseG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for n from 0 to 21 do" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 28 "numa:=convert(alpha,string):" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 59 "tracka[n]:=textplot([2.2,0.8,\"alpha is \".numa]
,color=blue);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "numb:=convert(beta
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1 0 30 "density[n]:=plot(f(x),x=0..3):" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 48 "P[n]:=display(\{density[n],tracka[n],trackb[n]\}):" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "alpha:=ainitial+n*alpha_step:" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 27 "beta:=binitial+n*beta_step:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 51 "f(x):=piecewise(x>alpha and x<beta,1/(beta-alpha
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{MPLTEXT 1 0 104 "display([seq(P[n], n=0..21)],insequence=true,title=
\"Changing alpha and beta, with beta - alpha fixed.\");" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
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t of 0.4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "UniformCDF(alpha..beta,0.4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"\"%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 54 "Areas can also be found using the integra
tion command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "evalf(int(f(x),x=alpha..0.4));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"\"%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 65 "a:=plot(f(x),x=alpha..beta,color=black,title=\"Uniform(0,1) PDF
\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "b:=plot(f(x),x=0.4.
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1 0 15 "display([a,b]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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71 "Find the value of the yellow shaded area, the area between 0.4 and
 0.6." }}{PARA 0 "" 0 "" {TEXT -1 97 "Using the CDF, we find the area \+
to the left of 0.6 and then subtract the area to the left of 0.4." }}
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{MPLTEXT 1 0 35 "area2:=UniformCDF(alpha..beta,0.4);" }}{PARA 11 "" 1 
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{MPLTEXT 1 0 26 "area_between:=area1-area2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%-area_betweenG$\"\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Using the integration com
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" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eval
f(int(f(x),x=0.4..0.6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"#!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "a:=plot(f(x),x=alpha.
.beta,color=black,title=\"Uniform(0,1) PDF\",labels=[\" \",\" \"]):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "b:=plot(f(x),x=0.75..beta,
color=yellow,filled=true,labels=[\" \",\" \"]):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "c:=textplot([0.75,-0.1,\"x\"],color=blue):" }}
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{TEXT -1 18 "Find the value of " }{TEXT 266 1 "x" }{TEXT -1 3 ".  " }
{TEXT 267 2 "x " }{TEXT -1 65 "is known as the third quartile for the \+
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*FenF-7$FailF(7&FcilF`il7$$\"1&)*ez]-$)***FenF-7$FfilF(7&FhilFeilFbxFb
x7\"-%&STYLEG6#%,PATCHNOGRIDG-Fdx6&Ffx$\"*++++\"!\")FajlF(-%%TEXTG6%7$
$\"#v!\"#$!\"\"F\\[m%\"xG-Fdx6&FfxF(F(Fajl-Fejl6$7$$\"\")F\\[m$\"\"#F
\\[m%%0.25G-%+AXESLABELSG6$Q\"~6\"F[\\m-%&TITLEG6#Q1Uniform(0,1)~PDF6
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45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 82 "Now we know the area under the curve to the left is 0.7
5 and we want to solve for " }{TEXT 268 1 "x" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f
solve(UniformCDF(alpha..beta,x)=0.75,x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+++++v!#5" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Propertie
s of the Distribution" }}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }
{TEXT 263 32 "Calculation of Mean and Variance" }}{PARA 0 "" 0 "" 
{TEXT -1 81 "To calculate the mean of the Uniform distribution,  we ca
n simply integrate over " }{TEXT 264 1 "x" }{TEXT -1 65 "*(Uniform PDF
),  per the definition of mathematical expectation. " }}{PARA 0 "" 0 "
" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta
rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "with(plots, display)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f(x):=UniformPDF(alpha
..beta,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#%\"xG*&\"\"\"F)
,&%%betaG\"\"\"%&alphaG!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "EX:=simplify(int(x*f(x),x=alpha..beta));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#EXG,&%%betaG#\"\"\"\"\"#%&alphaGF'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "No
tice that E(" }{TEXT 271 1 "X" }{TEXT -1 35 ") is midway between alpha
 and beta." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Calculating Var(" }
{TEXT 257 1 "X" }{TEXT -1 36 "), we will employ the formula:  Var(" }
{TEXT 258 1 "X" }{TEXT -1 7 ") = E( " }{XPPEDIT 18 0 "X^2;" "6#*$%\"XG
\"\"#" }{TEXT -1 5 " ) - " }{XPPEDIT 18 0 "( E(X) )^2" "6#*$-%\"EG6#%
\"XG\"\"#" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 38 "E_X_SQ:=int((x^2)*f(x),x=alpha..beta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'E_X_SQG,$*&,&*$)%%betaG\"\"$\"\"\"
\"\"\"*$)%&alphaGF+F,!\"\"F,,&F*F-F0F1!\"\"#F-F+" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 28 "VarX:=simplify(E_X_SQ-EX^2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%VarXG,(*$)%%betaG\"\"#\"\"\"#\"\"\"\"#7*&F(F,%&
alphaGF,#!\"\"\"\"'*$)F/F)F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 16 "Notice that Var(" }{TEXT 272 1 "X" }
{TEXT -1 16 ") simplifies to " }{XPPEDIT 18 0 "(beta-alpha)^2/12" "6#*
&,&%%betaG\"\"\"%&alphaG!\"\"\"\"#\"#7F(" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Simulati
on" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
281 "We will now simulate some data from a Uniform distribution in ord
er to compare the sampling distribution to the theoretical distributio
n.  The next bit of code will simulate a random sample of size 500 fro
m a Uniform(0,1) distribution, but you are invited to change the param
eters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "alpha:=0: beta:=1: n:=500:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 33 "sample:=UniformS(alpha..beta, n):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "Remember
 that when you plot histograms, the number of bins that you select can
 make a huge difference in the plot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "num_bins:=20:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a:=Histogram(sample,0..1,num_bins):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "b:=plot(UniformPDF(alph
a..beta,x),x=alpha..beta,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "display([a,b],title=\"Simulated vs. Theoretical\");" 
}}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "No
w let's compare the simulated mean to the theoretical mean.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 58 "theo_mean:=int(UniformPDF(alpha..beta,x)*x,x=alpha..beta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*theo_meanG#\"\"\"\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sim_mean:=Mean(sample);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%)sim_meanG$\"+9^T:\\!#5" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 46 "relative_error:=(sim_mean-theo_mean)/sim_
mean;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/relative_errorG$!++'33s\"!
#6" }}}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 }