On Mon, 28 Sep 2020 14:03:06 -0700 (PDT), RichD
<
r_delaney2001@yahoo.com> wrote:
On September 25, Rich Ulrich wrote:
I've been working through a pop math book of problems
and puzzles. The author is fond of the Poisson distribution.
The Poisson distribution arises from consideration of random,
independent, and "uniform" which has to be pretty basic.
?
Can you elaborate on that?
Not very well. You want to hear from someone who taught
Probablity 100. I can give and example and define terms, a bit.
If there are 1000 squares, what is the chance that the first
one will drop in square S? Then, the second drop?
"Independent" means that the second drop is not influenced
by the first drop or share tendencies with it about where to drop.
"Uniform" means that each square has an equal chance, for
each drop.
The number of drops that end up in each of the 1000 cells
will vary "randomly", with an average of [Total/1000]. The
resulting distribution of counts is what is called Poisson.
With a small mean, there will be many cells with 0 drops
That's especially when Poisson is useful for estimating.
When the mean is high, the Poisson shape resembles the
Normal distribution pretty closely, considering that it still
consists of integers (not continuous).
The earliest simple method of generation of "random numbers"
made use of taking a computer number (integer, from -32K to
+32K) to get a "uniform distribution" of integer results across
that range. Each NEXT number was obtained by manipulating
the current number -- pulling out bits, multiplying by a large
prime number, what-have-you. If done cleverly enough, the
computer would generate a series of 64K numbers: "uniform",
precisely (not randomly) across the range, before returning to
the initial number (called the "seed").
This are called "pseudo-random" -- each number is, indeed,
predictable, exactly, from the previous number. And there
are other tests of "randomness" that will fail, so, these days,
folks use other than this one. Look up "linear congruential
generator" for more information.
One old method of "generating a random /normal/ number"
was to take the average of 12 random, Uniform numbers.
These days, they use inverse-transformations from the Uniform.
The time between two Poisson (across time) events is
distributioned Expontial. A collection of events, i.e., the
counts of grouped Poisson events, approaches Normal as
the mean gets larger.
?
I'd expect that time intervals are Poisson distributed.
Another pop author might be more fond of the Normal.
My recollection is that it pops up in bus stop frequency
examples. So, is every bus stop problem associated with
a unique Poisson dist.? Conversely, does every such dist.
represent a bus stop problem?
I never saw many bus stop problems. Actually, they came
up in Queue theory, which starts (often) with Poisson and
extends to complications other than Normal.
I mean, given a problem presented as "m events / unit time",
is that always modeled by Poisson?
That's what comes to mind to me ... for large counts ... of identical
events (with low counts)... which are independent ... and if the
Normal assumption is not more convenient (because the mean is large).
Oh, the Binomial comes in when there is a limited max count,
like counting coin-flips. For large Ns, it looks like the normal, too.
--
Rich Ulrich
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