BarChart y 1, y 2, makes a bar chart with bar lengths y 1, y 2,.

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1 In[]:= In[]:= In[3]:= In[4]:= In[5]:= Out[5]= r : Random ri : Random Integer rdice : Random Integer,, 6 disp : Export "t.ps",, "EPS" & list Table rdice, 0 5,, 4, 6,, 3,, 3, 4,, 6, 4, 6,,, 6, 6,, 3, In[6]:= In[7]:= Statistics`DataManipulation` General::obspkg : Statistics`DataManipulation` is now obsolete. The legacy version being loaded may conflict with current Mathematica functionality. See the Compatibility Guide for updating information. In[8]:= In[9]:= Out[9]= fr : Frequencies & fr list 5,, 3,, 3, 3, 3, 4,, 5, 5, 6 In[0]:=? BarChart BarChart y, y, makes a bar chart with bar lengths y, y,. BarChart, w i y i,,, w j y j,, makes a bar chart with bar features defined by the symbolic wrappers w k. BarChart data, data, makes a bar chart from multiple datasets data i. In[]:= Graphics`Graphics` General::obspkg : Graphics`Graphics` is now obsolete. The legacy version being loaded may conflict with current Mathematica functionality. See the Compatibility Guide for updating information. Histogram::shdw : Symbol Histogram appears in multiple contexts Graphics`Graphics`, System` ; definitions in context Graphics`Graphics` may shadow or be shadowed by other definitions. BarChart::shdw : Symbol BarChart appears in multiple contexts Graphics`Graphics`, System` ; definitions in context Graphics`Graphics` may shadow or be shadowed by other definitions. BarSpacing::shdw : Symbol BarSpacing appears in multiple contexts Graphics`Graphics`, System` ; definitions in context Graphics`Graphics` may shadow or be shadowed by other definitions. PieChart::shdw : Symbol PieChart appears in multiple contexts Graphics`Graphics`, System` ; definitions in context Graphics`Graphics` may shadow or be shadowed by other definitions.

2 new688_probab.nb In[]:= BarChart fr list histogram Out[]= 0 In[3]:= Out[3]= In[4]:= In[5]:= Out[5]= In[6]:= Mean list standard 67 0 mymean list_ : Plus list Length list mymean list 67 0 StandardDeviation list N Out[6]= In[7]:= Out[7]= In[8]:= Variance list Sqrt N Out[8]= In[9]:= In[0]:= Out[0]= In[]:= Out[]= In[]:= In[3]:= Out[3]= Sample Space Timing list Table rdice, ; 0.047, Null Union list sample space all distinct possibilities,, 3, 4, 5, 6,,...6 "simple events", any combinations "events" list, 4, 6, 8 ; list Table i, i, 5,, 3, 4, 5

3 new688_probab.nb 3 In[4]:= Out[4]= In[5]:= Out[5]= In[6]:= Out[6]= In[7]:= Out[7]= In[8]:= Out[8]= In[9]:= Out[9]= In[30]:= Out[30]= In[3]:= Out[3]= list list,, 3, 4, 5, 6, 8 or Union list, list,, 3, 4, 5, 6, 8 Join list, list, 4, 6, 8,,, 3, 4, 5 Intersection list, list, 4 or list list, 4 allist Table i, i, 0,, 3, 4, 5, 6, 7, 8, 9, 0 Complement allist, list, 3, 5, 7, 9, 0 Complement allist, list, list 7, 9, 0 In[3]:= PROBABILTY sec. 4.3 in KR. In[33]:= In[34]:= In[35]:= consider rolling of dice; "simple events" are,3,.., roll : rdice rdice roll a single "experiment Out[35]= 9 In[36]:= In[37]:= Out[37]= In[38]:= Out[38]= now try MANY experiments Timing list Table roll, ;.96, Null Timing flist fr list 0.5, 8 078,, 55 4, 3, , 4, 468, 5, 38 36, 6, 66 56, 7, , 8, 80, 9, , 0, ,, 7 8,

4 4 new688_probab.nb In[39]:= BarChart flist Out[39]= In[40]:= ll Length list N Out[40]=. 0 6 In[4]:= Out[4]= In[4]:= fflist Table flist i, ll, flist i, N, i, ,., , 3., , 4., 0.468, 5., , 6., , 7., , 8., 0.8, 9., , 0., ,., 0.078,. BarChart fflist, BarStyle Blue Out[4]= In[43]:= In[44]:= Out[44]= In[45]:= Out[45]= note "lucky seven" fflist N ,., , 3., , 4., 0.468, 5., , 6., , 7., , 8., 0.8, 9., , 0., ,., 0.078,. exact Table i 36., i, i, 6 exact probabilites ,, , 3, , 4, 0., 5, , 6, , 7

5 new688_probab.nb 5 In[46]:= Out[46]= In[47]:= Table fflist i, exact i,, i, , , , 0.003, , Note: error is small, although we had a million tries statistical experiments converge slowly to exact probabilities In[48]:= In[49]:= In[50]:= Out[50]= In[5]:= Out[5]= In[5]:= Permutations Permutations a, b, c a, b, c, a, c, b, b, a, c, b, c, a, c, a, b, c, b, a Binomial n, m Binomial n, m n m n m FullSimplify Out[5]= 0 In[53]:= In[54]:= Factorial n Gamma n FullSimplify Out[54]= 0 In[55]:= Series n, n, Infinity, Stirling formula Out[55]= Log n n O n Π n 6 Π n O 3 n In[56]:= stir n_ n n n Π n Out[56]= n n n n Π In[57]:= stir 00. Out[57]= In[58]:= 00 N Out[58]= In[59]:= In[60]:= In[6]:= cummulativeprobability function cumprob n_ : Sum fflist i,, i, n N 36 cumprob Out[6]=.008

6 6 new688_probab.nb In[6]:= Plot cumprob n, n,,, PlotStyle Blue, AxesLabel "n", "F" Out[6]= In[63]:= In[64]:= In[65]:= mean list_ : Sum list i, list i,, i, Length list mean fflist N again, "lucky 7" Out[65]= In[66]:= In[67]:= var list_ : Block Μ mean list, Sum list i, Μ ^ list i,, i, Length list var fflist N Out[67]= In[68]:= fcoin, 0,, Out[68]=, 0,, In[69]:= Out[69]= In[70]:= Out[70]= mean fcoin var fcoin 4 In[7]:= In[7]:= In[73]:= approximation by a Gauss distribution gauss list_ : Block Μ mean list, Σ Sqrt var list, Σ Sqrt Pi Exp x Μ Σ ^

7 new688_probab.nb 7 In[74]:= Out[74]= gauss fflist N x In[75]:= In[76]:= In[77]:= In[78]:= In[79]:= mybarplot list_ : Graphics Red, Table Line list i,, 0, list i,, list i,, i, Length list Show mybarplot fflist, AspectRatio Out[79]= In[80]:= Show Plot gauss fflist, x, 0,, mybarplot fflist Out[80]= 0.05 In[8]:= binlist n_, p_ : Table Binomial n, m p ^m p ^ n m, m, m, 0, n

8 8 new688_probab.nb In[8]:= binlist 3, Out[8]= 8, 0, 3 8,, 3 8,, 8, 3 In[83]:= In[84]:= longbin binlist 0, ; mean longbin Out[84]= 0 In[85]:= Show Plot gauss longbin, x, 0, 0, mybarplot longbin 0.5 Out[85]= In[86]:= In[87]:= biasbin binlist 0,.3 ; Show Plot gauss biasbin, x, 0, 0, mybarplot biasbin, PlotRange 0, Out[87]= In[88]:= biasbin binlist 3,.6 ;

9 new688_probab.nb 9 In[89]:= Show Plot gauss biasbin, x,, 5, mybarplot biasbin, PlotRange 0, Out[89]= In[90]:= In[9]:= Out[9]= In[9]:= Out[9]= In[93]:= Out[93]= In[94]:= Out[94]= In[95]:= Central limittheorem: we already verified that the discrete "binary distribution" 0, for a random variable X which is NON Gaussian will lead to a Gaussian for a random variable Z, which is a sum of of n elements Z X... X for a large n. We will now try a different, continuous uniformdistribution but with the same mean and average as the "binary one" f x_ L L Integrate f x, x, a, a i.e. a L for the same average and normalization a L Integrate f x x, x, L, L mean Integrate f x x ^, x, L, L L Solve 4, L this gives the spread with varianve of 4 Out[95]= L 3, L 3 In[96]:= a Sqrt 3 N; b Sqrt 3 N Out[96]= In[97]:= rr : Random Real, a, b

10 0 new688_probab.nb In[98]:= Table rr, 0 Out[98]= , , , , , , , , 0.396, In[99]:= Clear sumn ; sumn n_ : Sum rr, n In[00]:= In[0]:= In[0]:= In[03]:= Out[03]= In[04]:= Do aa i 0, i, 00 ; run : Block s Floor sumn 00, aa s aa s Timing Do run, ;.8, Null listaa Table aa i, i, 00 Out[04]= 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3,, 5, 4, 4, 39, 67, 95, 38, 00, 39, 340, 480, 53, 593, 679, 749, 796, 807, 795, 70, 59, 5, 4, 393, 55, 86, 30, 97, 5, 3, 8, 3, 5, 0,, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 In[05]:= In[06]:= In[07]:= Out[07]= In[08]:= normal x_ : Sqrt 00 4 Sqrt Pi Exp x 50 Sqrt 00 4 ^ normal x x 5 Π Integrate normal x, x, Infinity, Infinity Out[08]= In[09]:= In[0]:= plolist : ListPlot listaa 0 000, PlotStyle Blue, PointSize.05 plonorm : Plot normal x, x, 0, 00, PlotStyle Red

11 new688_probab.nb In[]:= Show plolist, plonorm, PlotRange 0, 80, 0,., AxesOrigin 0, Out[]= In[]:= In[3]:= more on normal distribution In[4]:= cumul z_ Sqrt Pi Integrate Exp u^, u, Infinity, z Out[4]= In[5]:= Erf z cumul Infinity Out[5]= In[6]:= Plot Sqrt Pi Exp z ^, cumul z, z, 3, 3, PlotStyle Red, Blue Out[6]= In[7]:= Out[7]= FindRoot cumul z, z, Chop z 0

12 new688_probab.nb In[8]:= Out[8]= FindRoot cumul z 0.9, z, z.855 In[9]:= In[0]:= In[]:= Gauss from binomial cgaus m_, n_ : stir n stir m stir n m ^ n cgaus n x, n Out[]= Π n x n n x n x n n n n x n n n x n n x n x n x n x In[]:= Sqrt n. x y Sqrt n Out[]= Π n n n y n n y n n y n n 3 n n n y n n y n n y n n n y n n n y n n n y In[3]:= FullSimplify, Assumptions n 0 && y 0 Out[3]= n n n n y n n n Π Π y n n y n y In[4]:= Limit, n Infinity Out[4]= In[5]:= y Π this is the gauss distribution In[6]:= Gammaand chi^ distributions In[7]:= Out[7]= In[8]:= fgam x_, t_ lamexp lam x lam x ^ t Gamma t lam x lam lam x t Gamma t Integrate fgam x, t, x, 0, Infinity, Assumptions t 0 && lam 0 Out[8]= In[9]:= Out[9]= Integrate x fgam x, t, x, 0, Infinity, Assumptions t 0 && lam 0 average t lam

13 new688_probab.nb 3 In[30]:= Out[30]= In[3]:= Integrate x ^ fgam x, t, x, 0, Infinity, Assumptions t 0 && lam 0 variance t lam chi x_, n_ fgam x, n. lam Out[3]= In[3]:= n x x n Gamma n Plot chi x, 5, chi x, 0, x, 0, 0, PlotStyle Red, Blue, PlotRange 0, Out[3]=

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