Re: probability definition

2001-03-03 Thread Richard A. Beldin

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I'm glad to hear that somebody has his eye on the ball. Unfortunately, a
designation of a region like "western Puerto Rico" means so many
different things to so many different people, that I disbelieve its
utility. With the definition you quote, we should have a 100% chance of
precipitation almost every day.

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Re: probability definition

2001-03-02 Thread Richard A. Beldin

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The "definition" via axioms provides a mathematical structure that we
interpret either as "relative frequency" or as "degrees of belief".
Indeed, I think that any phenomenon which satisfies the axioms can serve
as an "interpretation". As they say, "If it walks like a duck, ...".

As far as the probability of rain tomorrow, I always explained to my
students that the language is so imprecise that the numerical value has
only rhetorical utility. We need to know:
1) How much rain in cm. ?
2) In which locations?
3) During what time span?

Does 70% probability mean that it rains in 70% of the locations or 70%
of the time or what?

Your instincts are correct. That example is severely flawed because we
have not made the experiment clear.

Continue to question the simple examples. You will learn from it.

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Re: probability definition

2001-03-02 Thread Harold E. Brooks

In article [EMAIL PROTECTED], "Richard A. Beldin"
[EMAIL PROTECTED] wrote:

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 The "definition" via axioms provides a mathematical structure that we
 interpret either as "relative frequency" or as "degrees of belief".
 Indeed, I think that any phenomenon which satisfies the axioms can serve
 as an "interpretation". As they say, "If it walks like a duck, ...".
 
 As far as the probability of rain tomorrow, I always explained to my
 students that the language is so imprecise that the numerical value has
 only rhetorical utility. We need to know:
 1) How much rain in cm. ?
 2) In which locations?
 3) During what time span?
 
 Does 70% probability mean that it rains in 70% of the locations or 70%
 of the time or what?
 
 Your instincts are correct. That example is severely flawed because we
 have not made the experiment clear.
 
 Continue to question the simple examples. You will learn from it.

Although in this case, in the US, the definition is quite precise for 
the National Weather Service.  From their Operations Manual, it is "the
likelihood of occurrence...of a precipitation event at any given point
in the forecast area.  The time period to which the PoP applies must
be clearly stated (or unambiguously inferred from the forecast wording)
since, without this, a numerical PoP value is meaningless."  Elsewhere
in the Ops Manual, a "precipitation event" is the occurrence of at least
0.01" of liquid equivalent precipitation (i.e., rain, melted snow).  It
has been shown that, given this definition, the PoP is equal to the 
expected areal coverage of the precipitation.  

Whether other groups issuing PoPs (media, other countries' weather 
services) follow the same definition, I don't know.At the locations
where verification data are available, NWS PoP forecasts out through
48 hours are remarkably reliable.  That is, if the PoP is N%, 
precipitation occurs very close to N% of the time.

Harold

-- 
[EMAIL PROTECTED]  http://www.nssl.noaa.gov/~brooks/
Standard disclaimer
Head, Mesoscale Applications Group, National Severe Storms Laboratory
Norman, Oklahoma


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Re: probability definition

2001-02-28 Thread Herman Rubin

In article [EMAIL PROTECTED],
James Ankeny [EMAIL PROTECTED] wrote:
 Hello,
   I have a question regarding the definition of probability. If I
understand correctly, probability may be defined using just axioms. However,
my textbook also uses a relative frequency definition, in which a
probability is defined as being the proportion of times an outcome occurs in
repeated trials of an experiment. This makes sense when one flip of the coin
is one trial, and in repeated trials, the proportion of heads is 1/2. But
what about a situation (an ex. in my textbook) where the probability of rain
tomorrow is 0.70. How do you define this experiment? Perhaps you measure
rainfall, temperature, pressure, etc. for each day over a long time period.
Then the probability of rain tomorrow is the proportion of times that rain
occurred on days with similar values for temp., humidity, etc.? This seems a
bit awkard to me. Also, how many trials must one perform an experiment,
before you know that the proportion converges to a particular fraction? Any
help on interpretation of relative frequency probabilities would be greatly
appreciated. In many cases, it seems difficult, at least for textbook
examples, to define what the actual experiment is. 


I think it is dangerous, and even useless, to ATTEMPT to
define probability.  In physics, one no longer even tries
to define length or mass, just specify their properties.


It is the same with probability.  A quantum mechanical
model has a joint probability distribution for observations,
but is worse between them.  Just as we use the postulated
properties for length and mass, we should use those for
probabilities.  We do have the nasty problem that there
is no way we can accurately calculate probabilities, unless
very strong additional assumptions are made.




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This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054   FAX: (765)494-0558


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