Re: What is the meaning of a single value in a continuous density function.
- From: Ken Butler <butler@xxxxxxxxxxxxxxxx>
- Date: Thu, 08 May 2008 11:50:16 -0400
On Thu, 8 May 2008 03:46:00 -0700 (PDT), sarikan
<serefarikan@xxxxxxxxx> wrote:
By theory, probability of a single value must be zero. However, you
can give a value to standard normal density function for example, and
get a number from R: dnorm(0,0,1,FALSE) gives 0.3989423
This is the probability at point 0 for standard normal distribution.
Not so: see my final paragraph.
The question is, although the function is capable of giving this
number, from a theory point of view, this number should be zero.
Would anyone care to comment on the meaning of this value? How do we
explain this contradiction? We have a value given by a function, but
it should be zero according to general rules of probability.
You can think of the definition of the density as being the derivative
of the cumulative distribution function (that is to say, the limit as
h tends to 0 of (F(x+h)-F(x))/h). This limit is (often) well defined,
eg. for the standard normal at x=0.
There are a couple of informal ways to interpret this:
1. how fast the cdf is changing (this comes directly from the
definition of the density as a derivative)
2. very roughly, "how likely you are to get a value near x", where the
definition of "near" comes from the limit. On a standard normal
distribution, the density is largest at 0, which means you are more
likely to get a value "near" 0 than near any other value. Because, in
the limit that defines the density, h is never actually 0, you are
always talking about the probability of an interval, which isn't
necessarily zero.
It isn't easy to see this, but maybe it helps to say that a density is
not itself a probability, so it won't behave like one.
Cheers,
Ken.
--
Ken Butler, Lecturer (Statistics)
University of Toronto at Scarborough
butler (at) utsc.utoronto.ca
http://www.utsc.utoronto.ca/~butler
.
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