Re: Distribution of a vowel on the page

On Sun, 21 Oct 2007 20:36:40 -0500, David Winsemius
<doe_snot@xxxxxxxxxxx> wrote:

Richard Ulrich <Rich.Ulrich@xxxxxxxxxxx> wrote in
news:t9snh31bag2npclbcrqq3s6ho30frufj0g@xxxxxxx:

On Sun, 21 Oct 2007 14:07:26 -0700, luca.pamparana@xxxxxxxxx wrote:

Hi everyone,

I have a statistics question. We were doing Poisson distribution in
our class today and the discussion topic is whether the number of
times a vowel occurs on a line follows Poisson distribution or not. I
think that it will not follow Poisson distribution as the probability
of a vowel occurring is more than the probability for a consonant
occurring. However, most people are of the opinion that it should
follow Poisson distribution because regardless of the probability it
should still be distributed randomly.

I did some test and took a page of paper from a normal book and did
some stats on it. I did not get it to look like Poisson distribution.
I am wondering which reasoning here is more reasonable.

In a book in English?

There are two reasons that the number of vowels on a line
will not follow Poisson. First, Poisson describes a relatively
*rare* event when the circumstance is otherwise binomial
(like this). Second, the events would have to be *independent*.

It is true that the Poisson is a good approximation to the binomial when
the rate parmeter is small, but the Poisson distribution may also be good
when even when the rate parmeter is not small. The question is really only
answerable by reference to the data.

I get it, if you are talking about the Poisson rate parameter
by itself, since that can be an arbitrary counter.
- Can you show me where a *binomial* rate parameter
is near or above 50% and results in Poisson appearance?


But consonants and vowels are structured by words, and
words seldom start or end with more than 3 (say) consonants
or vowels. What you observe will have almost no instances
of more than 6 -- of *either* consonants or vowels. And
one of those has p > 0.5, so the event should not be enormously
rare.

The average number of vowels per line must be around 20-25, so your
statement about "almost no instances of more than six" does not make much
sense to me. The next sentence makes even less sense. I am guessing that

Ah. I needed to be more clear. I was going on about the
*dependency*. If the occurrences are independent,
then there will be no "correlation" between *consecutive*
occurrences.

Clearly, vowels and consonants in words violate that assumption.
If there were a positive correlation (consecutive repetitions),
the distribution, if otherwise Poisson, would be over-dispersed
(variance too large for the mean). Since there is a negative
correlation, it will be under-dispersed.

It occurs to me that if there is a positive correlation of having
words with many-versus-few vowels, then the between-word r
might tend to offset the negative correlation within words.
I don't know how that would work out.
But the OP already stated that the empirical distribution
he obtained did not look Poisson.


you took the OP to be asking about was the number of one _specified_ vowel
and were not seeing the counts as _per_line_ but per word. At a rate of 25
vowels per line the Poisson is going to be distributed pretty much as a
Normal variable with mean=25 and s.d.=5. You should let the data determine
whether the added flexibility of a two parameter distribution like the
Normal is needed. I think the OP should provide his data and we can look it
over.

--
Rich Ulrich, wpilib@xxxxxxxx
http://www.pitt.edu/~wpilib/index.html
.

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