Monday

Homework: section 8.1 numbers 8,9,10,12 and section 8.2 numbers 8,10,11,13.

Note that the next exam will be handed out on Wednesday. Comments Here is the program from class used to explore standardized sums.

with(plots):

Areabargraph:=proc(data,xmin,xmax,k)

local sorteddata, dx, lines,graphicslist, f: sorteddata:=[]: dx:=(xmax-xmin)/k: lines:=[]: graphicslist:=[]: sorteddata:=sort(data): if ((op(1,sorteddata)<xmin) or (op(nops(sorteddata),sorteddata)>xmax)) then lprint(`Note: some data values lie outside the user-defined interval.`) fi: f:=proc(k,sorteddata,xmin,dx,xmax,lines::evaln) local i,j,currentupperlim,leng,index,counter,result, finallist,q,numb,linelist: finallist:=[]: linelist:=[]: index:=1: leng:=nops(sorteddata): currentupperlim:=xmin+dx: result:=[]: for q from 1 to leng do if ((op(q,sorteddata)>=xmin) and (op(q,sorteddata)<=xmax)) then finallist:=[op(finallist),op(q,sorteddata)] fi: od: numb:=nops(finallist): for i from 1 to k do counter:=0: while ((index<=numb) and (op(index,finallist)<=currentupperlim)) do counter:=counter+1: index:=index+1: od: result:=[op(result),counter]: currentupperlim:=currentupperlim+dx: od: for j from 1 to k do linelist:=[op(linelist),[xmin + (j-1)*dx,0]]: linelist:=[op(linelist),[xmin + (j-1)*dx,op(j,result)/(leng*dx)]]: linelist:=[op(linelist),[xmin + j*dx,op(j,result)/(leng*dx)]]: linelist:=[op(linelist),[xmin + j*dx,0]]: od: lines:=linelist: end: f(k,sorteddata,xmin,dx,xmax,lines): plot(lines,style=LINE); end:

"Here we plug in the function f of the uniform random variable on the unit interval , the number of times n we will average an experiment with this distribution, m the numbers of trials of this average, and the mean and variance of the the random variable whose inverse distribution is f. We get out a comparison of the standadr normal to the normalized sum.";

Ssum := proc(f,n,m,u,v)

local trans, U,V, explist, i , j, l, sum,standard,avlist,xmin,xmax: with(stats): avlist:=[]; for l from 1 to m do; explist:=[]; for i from 1 to n do; trans:=unapply(f,x): U := random[uniform[0,1]](); explist := [op(explist), trans(U)];

od;

sum:=0; for j from 1 to n do sum := sum + explist[j];

od;

standard := evalf((sum - n*u)/(sqrt(v*n)));

avlist := [op(avlist),standard]; od; xmax := max(op(avlist)); xmin:= min(op(avlist)); U := Areabargraph(avlist,xmin - .5,xmax + .5,30);

V := plot((1/(sqrt(2*Pi)))*exp(-x^2/2),x = xmin - .5 .. xmax + .5); display(U,V);

end:

"The uniform example";

Ssum(x,100,100,1/2,1/12);

Wednesday:

The only homework is the exam due by this Friday. Here is a list of typos and some clarifications in response to questions I received.

1.

(Actual typo!!!) In problem 3 imaging should be imagine.

2.

(Actual typo!!!!) In 1(b) I want you to figure out the varience of your winning in problem 1(a) (which you would have perhaps done any how to solve part 1(c), so in principle this makes you life easier).

3.

(Clarification.) The game in 1(d) is much simpler than the game in 1(a). If you play the game you receive (N_t) ! dollars. We view it as a game by imagining how much you should be willing to pay inorder to play it!

4.

(Clarification) In 1(e) the question is why is the constructed game is analogous to the Saint Petersberg paradox which we've played with and is explained in the text.

5.

(Clarification) In 3(d) indeed the programs have "frills" that can ignored when answering the question. For example you may simply explain the sense in which the distribtion produced by Weak(f(x),1,n) from week 6 is indeed capturing the the distribution of the random variable f(X) with X uniformly distributed on the unit interval.

with(plots):

E1 := plot(exp(-x),x=0..10): E2 := plot((exp(-x)*x)/(1),x=0..10): E3 := plot((exp(-x)*x^2)/(2!),x=0..10): E4 := plot((exp(-x)*x^3)/(3!),x=0..10): E5 := plot((exp(-x)*x^4)/(4!),x=0..10): display(E1,E2,E3,E4,E5);

A1 := plot(exp(-x),x=0..4): A2 := plot((2*exp(-2*x)*2*x)/(1),x=0..4): A3 := plot(6*(exp(-6*x)*(6*x)^5)/(5!),x=0..4): A4 := plot(28*(exp(-28*x)*(28*x)^27)/(27!),x=0..4): A5 := plot(18*(exp(-18*x)*(18*x)^17)/(17!),x=0..4): display(A1,A2,A3,A4,A5);

S3 := plot(sqrt(26)*(exp(-(sqrt(26)*x + 26))*(sqrt(26)*x + 26)^25)/(25!),x=-3..3): S4 := plot(sqrt(8)*(exp(-(sqrt(8)*x + 8))*(sqrt(8)*x + 8)^7)/(7!),x=-3..3): S5 := plot(sqrt(4)*(exp(-(sqrt(4)*x + 4))*(sqrt(4)*x + 4)^3)/(3!),x=-2..3): S2:= plot(sqrt(14)*(exp(-(sqrt(14)*x + 14))*(sqrt(14)*x + 14)^13)/(13!),x=-3..3): S1 := plot(sqrt(2)*(exp(-(sqrt(2)*x + 2))*(sqrt(2)*x + 2))/(1!),x=-.5..3): display(S1,S2,S3,S4,S5);

N1 := plot((sqrt(1)/(sqrt(2*Pi)))*exp(-(x^2)/2),x = -1..1): N2 := plot((sqrt(50)/(sqrt(2*Pi)))*exp(-(50*x^2)/2),x = -1..1): N3 := plot((sqrt(100)/(sqrt(2*Pi)))*exp(-(100*x^2)/2),x = -1..1): N4 := plot((sqrt(200)/(sqrt(2*Pi)))*exp(-(200*x^2)/2),x = -1..1): N5 := plot((sqrt(400)/(sqrt(2*Pi)))*exp(-(400*x^2)/2),x = -1..1): display(N1,N2,N3,N4,N5);

simplify(int((n/(pi^2))*(1/((x-t)^2+ 1))*(1/(x^2 + abs(n^2))),x = -infinity..infinity));

simplify(-I*n*Pi*(I+3*I*t^2+3*I*t^4-2*I*t^2*n^2+3*I*n^4-3*I*n^2+I*(n)*t^6+4*I*(n)^3*t^4-I*(n)+3*I*(n)*n^2+I*(n)*t^4-4*I*(n)^3*t^2+I*(n)*n^6+I*t^4*n^2-I*t^2*n^4+8*(n)^3*t^3+2*I*(n)*t^2*n^2-I*(n)*t^2*n^4-I*(n)*t^4*n^2+4*I*(n)^3*n^2*t^2-I*t^2*(n)-8*(n)*t^3*n^2-3*I*(n)*n^4-I*n^6+I*t^6)/(pi^2*(1+2*t^2-2*n^2+t^4+2*t^2*n^2+n^4)*(n^4-2*t^2*n^2-2*n^2+t^4+2*t^2+1+4*t^2*(n)^2)*(n)));

Friday

Homework

Chapter 7.1 numbers 3,5 and 6; and chapter 7.2 numbers 4,5 and 10.

Comments Here are the programs for chapter 7. Computer Demos.

Here are the program explored during fridays lecture

ConsolidateList:=proc(inputlist) local i,outputlist,temp,counter,templist,storelist,current,remember: outputlist:=[]: templist:=[]: storelist:=inputlist: counter:=0: while nops(templist)<nops(inputlist) do current:=op(1,storelist): remember:=1: for i from 1 to nops(storelist) do if (op(1,op(i,storelist))<op(1,current)) or (op(1,op(i,storelist))=op(1,current) and op(2,op(i,storelist))<op(2,current)) then current:=op(i,storelist): remember:=i: fi: od: templist:=[op(templist),current]: storelist:=subsop(remember=NULL,storelist): od: temp:=op(1,op(1,templist)): for i from 1 to nops(inputlist) do if op(1,op(i,templist))=temp and i<nops(templist) then counter:=counter+op(2,op(i,templist)) elif op(1,op(i,templist))=temp and i=nops(templist) then counter:=counter+op(2,op(i,templist)): outputlist:=[op(outputlist),[temp,counter]] elif op(1,op(i,templist))<>temp and i<nops(templist) then outputlist:=[op(outputlist),[temp,counter]]: temp:=op(1,op(i,templist)): counter:=op(2,op(i,templist)) else outputlist:=[op(outputlist),[temp,counter]]: outputlist:=[op(outputlist),op(i,templist)]: fi: od: outputlist; end:

Convolution:=proc(distributionlist1,distributionlist2) local i,j,outputlist,temp1,temp2: outputlist:=[]: for i from 1 to nops(distributionlist1) do for j from 1 to nops(distributionlist2) do temp1:=op(1,op(i,distributionlist1)) + op(1,op(j,distributionlist2)): temp2:=op(2,op(i,distributionlist1)) * op(2,op(j,distributionlist2)): outputlist:=[op(outputlist),[temp1,temp2]]: od: od: outputlist:=ConsolidateList(outputlist): end:

NFoldConvolution:=proc(distributionlist,n) local k,currentlist: currentlist:=distributionlist: for k from 2 to n do currentlist:=Convolution(currentlist,distributionlist): od: end:

SpikeData:=proc(data) local sorteddata,i,olist,plist,current,j,temp,distributionlist: olist:=sort(convert(convert(data,set),list)): sorteddata:=sort(data): plist:=[seq(0,j=1..nops(olist))]: current:=1: for i from 1 to nops(data) do if op(i,sorteddata)=op(current,olist) then temp:=op(current,plist): plist:=subsop(current=temp+1,plist): else current:=current+1: temp:=op(current,plist): plist:=subsop(current=temp+1,plist) fi: od: plist:=map(proc(x,data) evalf(x/nops(data)) end,plist,data): distributionlist:=zip(proc(x,y) [x,y] end,olist,plist): distributionlist: end:

Spikegraph:=proc(distributionlist,xmin,xmax,show) local num,linelist,j,i,templist: templist:=[]: if ((distributionlist[1][1]<xmin) or (distributionlist[nops(distributionlist)][1]>xmax) ) then lprint(`Note: some data values lie outside the user-defined interval.`) fi: for i from 1 to nops(distributionlist) do if not((distributionlist[i][1]<xmin) or (distributionlist[i][1]>xmax)) then templist:=[op(templist), [distributionlist[i][1], distributionlist[i][2] ] ]: fi: od: num:=nops(templist): linelist:=[]: for j from 1 to num do linelist:=[op(linelist),[templist[j][1],0]]: linelist:=[op(linelist),[templist[j][1],templist[j][2]]]: linelist:=[op(linelist),[templist[j][1],0]]: od: if show then plot(linelist,style=LINE) fi; end:

Summ := proc(n) local i,new,U; U := NFoldConvolution([[1,1/6],[2,1/6],[3,1/6],[4,1/6],[5,1/6],[6,1/6]],n); new := []; for i from 1 to 6*n - n do new := [op(new), [U[i][1],U[i][2]]]; od; Spikegraph(new, n,6*n,true ); end:

Summ(10);

Weak := proc(n) local i,new,U; U := NFoldConvolution([[1,1/6],[2,1/6],[3,1/6],[4,1/6],[5,1/6],[6,1/6]],n); new := []; for i from 1 to 6*n - n do new := [op(new), [U[i][1]/n,n*U[i][2]]]; od; Spikegraph(new, 0,6,true ); end: Weak(10);

Central := proc(n) local i,new,U,X,Y; U := NFoldConvolution([[1,1/6],[2,1/6],[3,1/6],[4,1/6],[5,1/6],[6,1/6]],n); new := []; for i from 1 to 6*n - n do new := [op(new), [evalf(((U[i][1]- n*(3.5))/sqrt(n*(35/12))) ),evalf(sqrt(n*(35/12))*(U[i][2]))]]; od; new; X := Spikegraph(new, -6,6,true ); Y := plot((1/(sqrt(2*Pi)))*exp(-x^2/2),x = -6 .. 6); display(X,Y); end:

Central(10);