RANDOM NUMBERS with BD

Transcription

1 In[1]:= BINOMIAL In[2]:=? BinomialDistribution BinomialDistribution n,p represents a binomialdistribution withn trials and successprobability p. In[3]:= In[4]:= Out[4]= bd : BinomialDistribution 1, 2 & pdf PDF bd n, p, m 1 p m n p m Binomial n, m 0 m n In[5]:= Plot pdf. n 20, p 1 2, m Floor x, x, 0, Out[5]= 0.05 In[6]:= Sum pdf, m, 0, n, Assumptions n 0 Out[6]= 1 In[7]:= Sum m pdf, m, 0, n, Out[7]= Assumptions n 1 && n Integers &&p 1 && p 0 n p In[8]:= Sum m n p ^2 pdf, m, 0, n, Assumptions n 1 && n Integers && p 1 && p 0 Out[8]= n p p 2 In[9]:= RANDOM NUMBERS with BD

2 2 114_BinPoiGa.nb In[10]:= Out[10]= RandomVariate bd 20,.5, 4 8, 9, 11, 5 In[11]:= Out[11]= Timing data RandomVariate bd 20,.5, ; , Null In[12]:= Out[12]= In[13]:= Mean data N Out[13]= In[14]:= StandardDeviation data N Out[14]= In[15]:= Sqrt Out[15]= In[16]:= In[17]:= Out[17]= In[18]:= In[19]:= POISSON from BINOMIAL pdf 1 p m n p m Binomial n, m 0 m n now n Infty, p 0, with n p mbar const pdf FullSimplify pdf. p mbar n, Assumptions n 1 && mbar 0 && mbar n Out[19]= In[20]:= Out[20]= In[21]:= Out[21]= mbarm n n mbar n m n Binomial n, m 0 m n Limit pdf, n Infinity mbar mbar m Gamma 1 m m 0 Ppdf E^ mbar m mbar^ m mbar mbar m m

3 114_BinPoiGa.nb 3 In[22]:= Out[22]= 1 Sum Ppdf, m, 0,Infinity In[23]:= Out[23]= Sum m Ppdf, m, 0, Infinity mbar In[24]:= Out[24]= Sum m mbar ^2 Ppdf, m, 0, Infinity mbar In[25]:= Plot Ppdf. mbar 10, m Floor x, pdf. n 20, mbar 10, m Floor x, x, 0, 20, PlotStyle Black, Red, Dashed 0.15 Out[25]= 0.05 In[26]:= built in Poisson In[27]:= Out[27]= In[28]:= Out[28]= PDF PoissonDistribution mu, m mu mu m m m 0 Timing data RandomVariate PoissonDistribution 10, ; , Null In[29]:= Out[29]= Mean data In[30]:= N Out[30]=

4 4 114_BinPoiGa.nb In[31]:= Variance data N Out[31]= In[32]:= GAUSS, and gauss from Binomial In[33]:= Gpdf PDF NormalDistribution mu, sig, x Out[33]= mu x 2 2 sig 2 2Π sig In[34]:= Integrate Gpdf, x, Infinity,Infinity, Assumptions sig 0 Out[34]= 1 In[35]:= Out[35]= Integrate x Gpdf, x, Infinity, Infinity, Assumptions sig 0 mu In[36]:= Integrate x mu ^2 Gpdf, Out[36]= sig 2 x, Infinity, Infinity, Assumptions sig 0 In[37]:= In[38]:= In[39]:= now from bd mu mbar; var mbar 1 mbar n ; sig Sqrt var ; Plot Gpdf. mbar 10, n 20, pdf. n 20, mbar 10, m Floor x, x, 0, 20, PlotStyle Black, Red, Dashed 0.15 Out[39]= 0.05

5 114_BinPoiGa.nb 5 In[40]:= Plot Gpdf. mbar 10, n 20, pdf. n 20, mbar 10, m Floor x 1 2, x, 0, 20, PlotStyle Black, Red, Dashed 0.15 Out[40]= 0.05 In[41]:= In[42]:= Out[42]= Gauss from Poisson, mbar 1 Ppdf mbar mbar m m In[43]:= In[44]:= Out[44]= Clear mu, sig, y ; mu mbar; sig Sqrt mbar ; Solve y m mbar sig, m m mbar mbar y In[45]:= Out[45]= 1 m mbar mbar y In[46]:= Ppdf. Out[46]= mbar mbar mbar mbar y mbar mbar y In[47]:= Out[47]= sig mbar 1 mbar mbar mbar y 2 mbar mbar y In[48]:= Out[48]= Limit, mbar Infinity y2 2 2Π

6 6 114_BinPoiGa.nb In[49]:= Gpdf Out[49]= In[50]:= mbar x 2 2 mbar mbar 2Π Plot Gpdf. mbar 10, Ppdf. mbar 10, m Floor x, x, 0, 20, PlotStyle Black, Red, Dashed Out[50]= In[51]:= Show Table Plot Gpdf. mbar i, Ppdf. mbar i, m Floor x, x, 0, 25, PlotStyle Black, Red, Dashed, i, 5, 15, Out[51]= 0.05 In[52]:= In[53]:= 25 note: "good" gaus, only mbar is matched ADVANCED: cumulative for gauss and poisson

7 114_BinPoiGa.nb 7 In[54]:= Clear mu, sig ; PDF NormalDistribution mu, sig, x Out[54]= In[55]:= Out[55]= In[56]:= mu x 2 2 sig 2 2Π sig Integrate, x, Infinity, X, Assumptions X 0 && sig 0 1 mu X Erfc 2 2 sig Plot. mu 4, sig 1, X, 0, Out[56]= In[57]:= or, built in In[58]:= Out[58]= Fg x_ CDF NormalDistribution mu, sig, x 1 mu x Erfc 2 2 sig In[59]:= Out[59]= Fg mu sig Fg mu sig 1 2 Erfc Erfc 1 2 In[60]:= N Out[60]= In[61]:= Fg mu 2 sig Fg mu 2 sig N Out[61]= In[62]:= In[63]:= Out[63]= now poisson Ppdf mbar mbar m m

8 8 114_BinPoiGa.nb In[64]:= Sum, m, 0, M, Assumptions M 0 Out[64]= Gamma 1 M, mbar Gamma 1 M In[65]:= Plot. mbar 10, M Floor x, x, 0, Out[65]= In[66]:= In[67]:= Out[67]= In[68]:= In[69]:= Out[69]= or, built in CDF PoissonDistribution mu, m GammaRegularized 1 Floor m, mu m 0 ADVANCED: summation theorem for Poisson Timing data1 RandomVariate PoissonDistribution 5, ; , Null In[70]:= In[71]:= Out[71]= Timing data2 RandomVariate PoissonDistribution 10, ; , Null

9 114_BinPoiGa.nb 9 In[72]:= Show Histogram data1, data2, data1 data2, ChartStyle Orange, White, Blue, Plot Ppdf. m x 1 2, mbar 15, x, 0, Out[72]= In[73]:= Out[73]= Export "t.ps",, "EPS" t.ps In[74]:= In[75]:= In[76]:= 114_sumPoi.ps ADV.: adding 2 ND p1 Gpdf. x y, mu mu1, sig sig1 Out[76]= In[77]:= mu1 y 2 2 sig1 2 2Π sig1 p2 Gpdf. x x y, mu mu2, sig sig2 Out[77]= In[78]:= Out[78]= mu2 x y 2 2 sig2 2 2Π sig2 Integrate p1 p2, y, Infinity, Infinity, Assumptions sig1 0 && sig2 0 mu1 mu2 x 2 2 sig1 2 sig2 2 2Π sig1 2 sig2 2 In[79]:= Out[79]= Timing data1 RandomVariate NormalDistribution 3, 3, ; , Null In[80]:= Out[80]= Timing data2 RandomVariate NormalDistribution 7, 4, ; , Null In[81]:= Max data1 data2 Min data1 data2 Out[81]=

10 10 114_BinPoiGa.nb In[82]:= Show Histogram data1, data2, data1 data2, 50, ChartStyle Green, White, Red, Plot Gpdf. m x 1 2, mu 10, sig 5, x, 0, Out[82]= In[83]:= Out[83]= Export "t.ps",, "EPS" t.ps In[84]:= In[85]:= In[86]:= Out[86]= 114_addGa.ps CAUCHY Lorenz distribution Cpdf PDF CauchyDistribution mu, gam, x Simplify gam Π gam 2 mu x 2 In[87]:= Out[87]= 1 Integrate Cpdf, x, Infinity, Infinity, Assumptions gam 0 && mu 0 In[88]:= Out[88]= In[89]:= In[90]:= Integrate Cpdf, x, Infinity, mu, Assumptions gam 0 && mu xbar does not exist Plot Cpdf. mu 2, gam.05, x, 1, 5, PlotRange 0, 1.05 Pi Out[90]=

11 114_BinPoiGa.nb 11 In[91]:= In[92]:= Out[92]= In[93]:= CDF CauchyDistribution mu, gam, x Simplify 1 2 ArcTan mu x gam Π Plot. mu 2, gam.05, x, 1, 5, PlotRange 0, Out[93]=