density vs. mass for discrete probability functions

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density vs. mass for discrete probability functions

Stefan Schreiber
Dear R users,

While experimenting with the dbinom() function and reading its
documentation (?dbinom) it reads that "dbinom gives the density" but
shouldn't it be called "mass" instead of "density"? I assume that it
has something to do with keeping the function for "density" consistent
across discrete and continuous probability functions - but I am not
sure and was hoping someone could clarify?

Furthermore the help file for dbinom() function references a link
(http://www.herine.net/stat/software/dbinom.html) but it doesn't seem
to land where it should. Maybe this could be updated?

Thank you,
Stefan

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Re: density vs. mass for discrete probability functions

Spencer Graves-4


On 2019-03-14 19:43, Stefan Schreiber wrote:
> Dear R users,
>
> While experimenting with the dbinom() function and reading its
> documentation (?dbinom) it reads that "dbinom gives the density" but
> shouldn't it be called "mass" instead of "density"? I assume that it
> has something to do with keeping the function for "density" consistent
> across discrete and continuous probability functions - but I am not
> sure and was hoping someone could clarify?


       The Wikipedia article on "Probability density function" gives the
"Formal definition" that, "the density of [a random variable] with
respect to a reference measure ... is the Radon–Nikodym derivative".


       This sounds bazaar to people who haven't studied
measure-theoretic probability, but it allows a unified treatment of
continuous and discrete probabilities and to others that are
combinations and neither.  The "reference measure" for a discrete
probability distribution is the "counting measure", which supports the
use of the word "density" in this context being equivalent to "mass". 
For continuous distributions, the "reference measure" is routinely taken
to be the "improper prior" that assigns measure 1 to any unit interval
on the real line.


       Does that make it clear as mud?


       Spencer Graves


https://en.wikipedia.org/wiki/Probability_density_function

>
> Furthermore the help file for dbinom() function references a link
> (http://www.herine.net/stat/software/dbinom.html) but it doesn't seem
> to land where it should. Maybe this could be updated?
>
> Thank you,
> Stefan
>
> ______________________________________________
> [hidden email] 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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Re: density vs. mass for discrete probability functions

Peter Dalgaard-2
In reply to this post by Stefan Schreiber
Mathematically, you can bring discrete and continuous distributions on a common footing by defining probability functions as densities wrt. counting measure. You don't really need Radon-Nikodym derivatives to understand the idea, just the fact that sums can be interpreted as integrals wrt counting measure, hence sum_{x in A} f(x) and int_A f(x) dx are essentially the same concept.

-pd

> On 15 Mar 2019, at 01:43 , Stefan Schreiber <[hidden email]> wrote:
>
> Dear R users,
>
> While experimenting with the dbinom() function and reading its
> documentation (?dbinom) it reads that "dbinom gives the density" but
> shouldn't it be called "mass" instead of "density"? I assume that it
> has something to do with keeping the function for "density" consistent
> across discrete and continuous probability functions - but I am not
> sure and was hoping someone could clarify?
>
> Furthermore the help file for dbinom() function references a link
> (http://www.herine.net/stat/software/dbinom.html) but it doesn't seem
> to land where it should. Maybe this could be updated?
>
> Thank you,
> Stefan
>
> ______________________________________________
> [hidden email] 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.

--
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Office: A 4.23
Email: [hidden email]  Priv: [hidden email]

______________________________________________
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Re: density vs. mass for discrete probability functions

Stefan Schreiber
Thank you Peter and Spencer. That clears things up. Also since no one
responded the second part of my question, I'm still wondering if it was
noted that there is a hyperlink in the dbinom help file (?dbinom) that
isn't directing correctly?

Stefan

On Fri, Mar 15, 2019, 07:37 peter dalgaard, <[hidden email]> wrote:

> Mathematically, you can bring discrete and continuous distributions on a
> common footing by defining probability functions as densities wrt. counting
> measure. You don't really need Radon-Nikodym derivatives to understand the
> idea, just the fact that sums can be interpreted as integrals wrt counting
> measure, hence sum_{x in A} f(x) and int_A f(x) dx are essentially the same
> concept.
>
> -pd
>
> > On 15 Mar 2019, at 01:43 , Stefan Schreiber <[hidden email]>
> wrote:
> >
> > Dear R users,
> >
> > While experimenting with the dbinom() function and reading its
> > documentation (?dbinom) it reads that "dbinom gives the density" but
> > shouldn't it be called "mass" instead of "density"? I assume that it
> > has something to do with keeping the function for "density" consistent
> > across discrete and continuous probability functions - but I am not
> > sure and was hoping someone could clarify?
> >
> > Furthermore the help file for dbinom() function references a link
> > (http://www.herine.net/stat/software/dbinom.html) but it doesn't seem
> > to land where it should. Maybe this could be updated?
> >
> > Thank you,
> > Stefan
> >
> > ______________________________________________
> > [hidden email] 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.
>
> --
> Peter Dalgaard, Professor,
> Center for Statistics, Copenhagen Business School
> Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> Phone: (+45)38153501
> Office: A 4.23
> Email: [hidden email]  Priv: [hidden email]
>
>
>
>
>
>
>
>
>
>

        [[alternative HTML version deleted]]

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Re: density vs. mass for discrete probability functions

Spencer Graves-4
In reply to this post by Peter Dalgaard-2


On 2019-03-15 08:37, peter dalgaard wrote:
> Mathematically, you can bring discrete and continuous distributions on a common footing by defining probability functions as densities wrt. counting measure. You don't really need Radon-Nikodym derivatives to understand the idea, just the fact that sums can be interpreted as integrals wrt counting measure, hence sum_{x in A} f(x) and int_A f(x) dx are essentially the same concept.


       Correct.  That's for clearing up my "mud".  sg

> -pd
>
>> On 15 Mar 2019, at 01:43 , Stefan Schreiber <[hidden email]> wrote:
>>
>> Dear R users,
>>
>> While experimenting with the dbinom() function and reading its
>> documentation (?dbinom) it reads that "dbinom gives the density" but
>> shouldn't it be called "mass" instead of "density"? I assume that it
>> has something to do with keeping the function for "density" consistent
>> across discrete and continuous probability functions - but I am not
>> sure and was hoping someone could clarify?
>>
>> Furthermore the help file for dbinom() function references a link
>> (http://www.herine.net/stat/software/dbinom.html) but it doesn't seem
>> to land where it should. Maybe this could be updated?
>>
>> Thank you,
>> Stefan
>>
>> ______________________________________________
>> [hidden email] 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.

______________________________________________
[hidden email] 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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Re: density vs. mass for discrete probability functions

JLucke
In reply to this post by Stefan Schreiber
Stefan---

Under the measure-theoretic approach to probability, discrete & continuous
probability densities follow the same underlying mathematical principles.
Check any text on measure-theoretic probability theory.

---JFL





Stefan Schreiber <[hidden email]>
Sent by: "R-help" <[hidden email]>
03/14/2019 08:43 PM

To
[hidden email],
cc

Subject
[R] density vs. mass for discrete probability functions






Dear R users,

While experimenting with the dbinom() function and reading its
documentation (?dbinom) it reads that "dbinom gives the density" but
shouldn't it be called "mass" instead of "density"? I assume that it
has something to do with keeping the function for "density" consistent
across discrete and continuous probability functions - but I am not
sure and was hoping someone could clarify?

Furthermore the help file for dbinom() function references a link
(http://www.herine.net/stat/software/dbinom.html) but it doesn't seem
to land where it should. Maybe this could be updated?

Thank you,
Stefan

______________________________________________
[hidden email] 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]]

______________________________________________
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Re: density vs. mass for discrete probability functions

Rui Barradas
In reply to this post by Stefan Schreiber
Hello,

Yes, there is even an old discussion on this on r-devel, dated August,
10 2013.
See [1].


[1]
https://r-project.markmail.org/search/?q=broken-link-in-docs-for-Binormial-functions#query:broken-link-in-docs-for-Binormial-functions+page:1+mid:rf6tbiokcdyai6el+state:results


Hope this helps,



Rui Barradas

Às 14:21 de 15/03/2019, Stefan Schreiber escreveu:

> Thank you Peter and Spencer. That clears things up. Also since no one
> responded the second part of my question, I'm still wondering if it was
> noted that there is a hyperlink in the dbinom help file (?dbinom) that
> isn't directing correctly?
>
> Stefan
>
> On Fri, Mar 15, 2019, 07:37 peter dalgaard, <[hidden email]> wrote:
>
>> Mathematically, you can bring discrete and continuous distributions on a
>> common footing by defining probability functions as densities wrt. counting
>> measure. You don't really need Radon-Nikodym derivatives to understand the
>> idea, just the fact that sums can be interpreted as integrals wrt counting
>> measure, hence sum_{x in A} f(x) and int_A f(x) dx are essentially the same
>> concept.
>>
>> -pd
>>
>>> On 15 Mar 2019, at 01:43 , Stefan Schreiber <[hidden email]>
>> wrote:
>>>
>>> Dear R users,
>>>
>>> While experimenting with the dbinom() function and reading its
>>> documentation (?dbinom) it reads that "dbinom gives the density" but
>>> shouldn't it be called "mass" instead of "density"? I assume that it
>>> has something to do with keeping the function for "density" consistent
>>> across discrete and continuous probability functions - but I am not
>>> sure and was hoping someone could clarify?
>>>
>>> Furthermore the help file for dbinom() function references a link
>>> (http://www.herine.net/stat/software/dbinom.html) but it doesn't seem
>>> to land where it should. Maybe this could be updated?
>>>
>>> Thank you,
>>> Stefan
>>>
>>> ______________________________________________
>>> [hidden email] 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.
>>
>> --
>> Peter Dalgaard, Professor,
>> Center for Statistics, Copenhagen Business School
>> Solbjerg Plads 3, 2000 Frederiksberg, Denmark
>> Phone: (+45)38153501
>> Office: A 4.23
>> Email: [hidden email]  Priv: [hidden email]
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>
> [[alternative HTML version deleted]]
>
> ______________________________________________
> [hidden email] 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.
>

______________________________________________
[hidden email] 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.