; This example illustrates the ubiquity of random variables.
; At the same time that I was finalizing this Examples folder,
; I was reading a novel from our local public library.  Like
; most novels, the text was right-justified, in this case,
; with hyphens at the end of many lines.  The number of hyphens
; per page, Y, is a random variable.

; There were N = 18 lines per page, on average, that were long enough 
; to have a hyphen.  If you make the reasonable assumption that 
; the probability of a hyphen at the end of any line, p, is a constant
; and independent of other lines, then Y should be ~Binomial(p, 18).
; Is it?

; A classic rule-of-thumb says that if N > 49 and Np < 5, then the
; event in question qualifies as a "rare" event.  If so, then the
; distribution might be ~Poisson(Np).  The presence of a hyphen would 
; not seem to qualify as rare.  Still, there is more than one reason why 
; a variable might be ~Poisson.  Why not try it?

; Here is the observed distribution of hyphens per page:

@0 20
@1 77
@2 93
@3 83
@4 48
@5 32
@6 20
@7 3
@8 2

Comments are prefixed with a semicolon. The discrete data are grouped, with bin values prefixed by @, followed by the bin frequency. The density function shown is a Poisson distribution. Like all examples shown here, this model was found to be "acceptable" (probability > 0.1), based on the worse of two measured goodness-of-fit metrics, here maximum-likelihood (ML), which was also the (user-selected) optimization criterion, and Chi-squared.

Model

Parameter

• A -- expectation

The ML Binomial model, pictured below, is deemed to be "unacceptable" (probability < 0.05) but not "very unacceptable" (probability < 0.01) based on a parametric boostrap with 1,000 bootstrap samples synthesized from the optimum, ML, model.

This dataset is one of the examples included in the Regress+ software package.