LOG TO SUMMARIZE SOME SIMULATION AND GOODNESS OF FIT STEPS
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## START WITH SIMULATION & CHECKING

> unix.time({tmpdat = array(rexp(2e5), c(200,1000))})
   user  system elapsed 
   0.08    0.00    0.12 
  
  breakpt = c(0,1,2,4,50)
  pvec = diff(pexp(breakpt))
> round(pvec,5)
[1] 0.63212 0.23254 0.11702 0.01832

> nobs = hist(tmpdat[,1], breaks=breakpt, plot=F)$count
> nobs
[1] 118  56  24   2
> sum((nobs-200*pvec)^2/(200*pvec))
[1] 3.268488

> chisq.test(nobs, p=pvec)

        Chi-squared test for given probabilities

data:  nobs 
X-squared = 3.2685, df = 3, p-value = 0.3521

Warning message:
In chisq.test(nobs, p = pvec) : Chi-squared approximation may be incorrect

> ChiTsts = apply(tmpdat,2, function(dcol) 
        chisq.test(hist(dcol, breaks=breakpt, 
                     plot=F)$count, p=pvec)$stat)
## Same warning message is repeated many times !

> length(ChiTsts)
[1] 1000

> summary(ChiTsts)
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
 0.05315  1.15500  2.33500  2.92600  4.02500 16.17000 
  #### Compare to chi-sq 3df distribution with mean 3.

### For a graphical picture (scaled relative frequency 
###       histogram with overlaid chisq_3 density):

> hist(ChiTsts, prob=T, nclass=40,
     main="Hist of Goodness of FitChisq vs Chisq_3")
  curve(dchisq(x,3), add=T, lty=3, col="red")

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### How many times would we expect  Chisq > 15 ?
> 1000*(1-pchisq(15,3))
[1] [1] 1.816649           ### p-value .00182 each time

> sum(ChiTsts > 15)
[1] 2                
   ### Distribution of the random variate just produced ?
   ### Approximately Poisson(1.816649)

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