[R] Ignoring the domain of RV in punif()

[Hamed Ha]

Hi Ted,

Thanks for the explanation.

I am convinced at least more than average by Eric and your answer. But
still have some shadows of confusion that is definitely because I have
forgotten some fundamentals in probabilities.

In your cube example, the cumulative probability of reaching a point
outside the cube (u or v or w  > A) is 1 however, the bigger cube does not
exists (because the Q is the reference space). Other words, I feel that we
extend the space to accommodate any cube of any size! Looks a bit weird to
me!


Hamed.

On Tue, 23 Oct 2018 at 11:52, Ted Harding <ted.harding using wlandres.net> wrote:

> Sorry -- stupid typos in my definition below!
> See at ===*** below.
>
> On Tue, 2018-10-23 at 11:41 +0100, Ted Harding wrote:
> Before the ticket finally enters the waste bin, I think it is
> necessary to explicitly explain what is meant by the "domain"
> of a random variable. This is not (though in special cases
> could be) the space of possible values of the random variable.
>
> Definition of (real-valued) Random Variable (RV):
> Let Z be a probability space, i.e. a set {z} of entities z
> on which a probability distribution is defined. The entities z
> do not need to be numeric. A real-valued RV X is a function
> X:Z --> R defined on Z such that, for any z in Z, X(z) is a
> real number. The set Z, in tthis context, is (by definitipon)
> the *domain* of X, i.e. the space on which X is defined.
> It may or may not be (and usually is not) the same as the set
> of possible values of X.
>
> Then. given any real value x0, the CDF of X at x- is Prob[X <= X0].
> The distribution function of X does not define the domain of X.
>
> As a simple exam[ple: Suppose Q is a cube of side A, consisting of
> points z=(u,v,w) with 0 <= u,v,w <= A. Z is the probability space
> of points z with a uniform distribution of position within Q.
> Define the random variable X:Q --> [0,1] as
> ===***
>   X[u,v,w) = x/A
>
> Wrong! That should  have been:
>
>   X[u,v,w) = w/A
> ===***
> Then X is uniformly distributed on [0,1], the domain of X is Q.
> Then for x <= 0 _Prob[X <= x] = 0, for 0 <= x <= 1 Prob(X >=x] = x,
> for x >= 1 Prob(X <= x] = 1. These define the CDF. The set of poaaible
> values of X is 1-dimensional, and is not the same as the domain of X,
> which is 3-dimensional.
>
> Hopiong this helps!
> Ted.
>
> On Tue, 2018-10-23 at 10:54 +0100, Hamed Ha wrote:
> > > Yes, now it makes more sense.
> > >
> > > Okay, I think that I am convinced and we can close this ticket.
> > >
> > > Thanks Eric.
> > > Regards,
> > > Hamed.
> > >
> > > On Tue, 23 Oct 2018 at 10:42, Eric Berger <ericjberger using gmail.com>
> wrote:
> > >
> > > > Hi Hamed,
> > > > That reference is sloppy. Try looking at
> > > > https://en.wikipedia.org/wiki/Cumulative_distribution_function
> > > > and in particular the first example which deals with a Unif[0,1] r.v.
> > > >
> > > > Best,
> > > > Eric
> > > >
> > > >
> > > > On Tue, Oct 23, 2018 at 12:35 PM Hamed Ha <hamedhaseli using gmail.com>
> wrote:
> > > >
> > > >> Hi Eric,
> > > >>
> > > >> Thank you for your reply.
> > > >>
> > > >> I should say that your justification makes sense to me.  However, I
> am in
> > > >> doubt that CDF defines by the Pr(x <= X) for all X? that is the
> domain of
> > > >> RV is totally ignored in the definition.
> > > >>
> > > >> It makes a conflict between the formula and the theoretical
> definition.
> > > >>
> > > >> Please see page 115 in
> > > >>
> > > >>
> https://books.google.co.uk/books?id=FEE8D1tRl30C&printsec=frontcover&dq=statistical+distribution&hl=en&sa=X&ved=0ahUKEwjp3PGZmJzeAhUQqxoKHV7OBJgQ6AEIKTAA#v=onepage&q=uniform&f=false
> > > >> The
> > > >>
> > > >>
> > > >> Thanks.
> > > >> Hamed.
> > > >>
> > > >>
> > > >>
> > > >> On Tue, 23 Oct 2018 at 10:21, Eric Berger <ericjberger using gmail.com>
> wrote:
> > > >>
> > > >>> Hi Hamed,
> > > >>> I disagree with your criticism.
> > > >>> For a random variable X
> > > >>> X: D - - - > R
> > > >>> its CDF F is defined by
> > > >>> F: R - - - > [0,1]
> > > >>> F(z) = Prob(X <= z)
> > > >>>
> > > >>> The fact that you wrote a convenient formula for the CDF
> > > >>> F(z) = (z-a)/(b-a)  a <= z <= b
> > > >>> in a particular range for z is your decision, and as you noted this
> > > >>> formula will give the wrong value for z outside the interval [a,b].
> > > >>> But the problem lies in your formula, not the definition of the CDF
> > > >>> which would be, in your case:
> > > >>>
> > > >>> F(z) = 0 if z <= a
> > > >>>        = (z-a)/(b-a)   if a <= z <= b
> > > >>>        = 1 if 1 <= z
> > > >>>
> > > >>> HTH,
> > > >>> Eric
> > > >>>
> > > >>>
> > > >>>
> > > >>>
> > > >>> On Tue, Oct 23, 2018 at 12:05 PM Hamed Ha <hamedhaseli using gmail.com>
> wrote:
> > > >>>
> > > >>>> Hi All,
> > > >>>>
> > > >>>> I recently discovered an interesting issue with the punif()
> function.
> > > >>>> Let
> > > >>>> X~Uiform[a,b] then the CDF is defined by F(x)=(x-a)/(b-a) for
> (a<= x<=
> > > >>>> b).
> > > >>>> The important fact here is the domain of the random variable X.
> Having
> > > >>>> said
> > > >>>> that, R returns CDF for any value in the real domain.
> > > >>>>
> > > >>>> I understand that one can justify this by extending the domain of
> X and
> > > >>>> assigning zero probabilities to the values outside the domain.
> However,
> > > >>>> theoretically, it is not true to return a value for the CDF
> outside the
> > > >>>> domain. Then I propose a patch to R function punif() to return an
> error
> > > >>>> in
> > > >>>> this situations.
> > > >>>>
> > > >>>> Example:
> > > >>>> > punif(10^10)
> > > >>>> [1] 1
> > > >>>>
> > > >>>>
> > > >>>> Regards,
> > > >>>> Hamed.
> > > >>>>
> > > >>>>         [[alternative HTML version deleted]]
> > > >>>>
> > > >>>> ______________________________________________
> > > >>>> R-help using r-project.org mailing list -- To UNSUBSCRIBE and more, see
> > > >>>> https://stat.ethz.ch/mailman/listinfo/r-help
> > > >>>> PLEASE do read the posting guide
> > > >>>> http://www.R-project.org/posting-guide.html
> > > >>>> and provide commented, minimal, self-contained, reproducible code.
> > > >>>>
> > > >>>
> > >
> > >     [[alternative HTML version deleted]]
> > >
> > > ______________________________________________
> > > R-help using r-project.org mailing list -- To UNSUBSCRIBE and more, see
> > > https://stat.ethz.ch/mailman/listinfo/r-help
> > > PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html
> > > and provide commented, minimal, self-contained, reproducible code.
> >
> > ______________________________________________
> > R-help using r-project.org mailing list -- To UNSUBSCRIBE and more, see
> > https://stat.ethz.ch/mailman/listinfo/r-help
> > PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html
> > and provide commented, minimal, self-contained, reproducible code.
>
>

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