Re: Probability in an infinite sample space

On Tue, 21 Mar 2006 00:14:00 +1100, Peter Webb wrote:

"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
news:dvl745$l1c$1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On 19 Mar 2006 19:04:41 -0800, mikeh106@xxxxxxxxxxx wrote:
If you choose a natural number at random, that it will be a multiple of
3 is there a 1/3, 1/2, or undefined chance?

That's a meaningless question, because you have not identified a
probability distribution.

It is often assumed that if no distribution is specified, then a uniform
distribution is intended. That can't be the case here, because there is
no such thing as a uniform probability distribution on a countably
infinite sample space.

Does it mean something to select "at random" from an infinite number of
objects?

Certainly, provided a probability distribution is specified. For
example, there is a uniform distribution on the unit interval [0,1],
which is an uncountably infinite space. You can also have a probability
distribution on a countably infinite space, but it can't be uniform. For
example, the probability might be P(n) = 2^(-n) for n = 1, 2, 3, ....


--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>

You and another poster strengthened "uniform distribution over N" to
"uniform distribution over countably infinite", so I realise that my
following remarks are almost certainly wrong.

Consider the set of reals [0,1] that has a terminating decimal
representation. Consider the same probability distribution over this as for
all Reals [0,1] uniformly. Isn't this a uniform distribution over a
countably infinite set?

It isn't a probability distribution. The total weight is 0, not 1.

Probability distributions are required to be countably additive, but they are
not *uncountably* additive.



--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.

Relevant Pages

  • Re: Probability in an infinite sample space
    ... It is often assumed that if no distribution is specified, then a uniform ... no such thing as a uniform probability distribution on a countably ... distribution on a countably infinite space, ...
    (sci.math)
  • Re: An uncountable countable set
    ... there are no uniform distributions over countable sample spaces, ... probability distribution. ... probability associated with it: that fraction. ...
    (sci.math)
  • Re: Sort benchmark and a classic paper by Gray
    ... are sufficiently random. ... random" for a binary value to mean a uniform (flat) probability distribution. ... If all the bits are indeed random, the resulting distribution is of course uniform. ...
    (comp.arch)
  • Re: Compression contests
    ... whether the next symbol's probability distribution is affected by the ... your algorithm on so called "memoryless sources" with an "uniform ... The terms "memoryless source" should be just as alarming as "uniform ... can't predict the sequence no matter what algorithm you use. ...
    (comp.compression)
  • Re: Confirmation of Shannons Mistake about Perfect Secrecy of One-time-pad
    ... of M from the known joint probability distribution ... probability distribution of M and C when C is fixed) ... the assumption that K is uniform when C is fixed ... is incorrect. ...
    (sci.math)
Loading