On Thu, 8 May 2008 03:46:00 -0700 (PDT), sarikan
<serefarikan@[EMAIL PROTECTED]
> 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 ***ulative 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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