# [R] median and joint distribution

Hotz, T. th50 at leicester.ac.uk
Thu Jul 24 14:42:42 CEST 2003

```Dear Salvatore,

Assuming that you mean "convolution" when you write
answer. It will depend heavily on the joint distribution
of the two random variables.

Just to give a simple example, let X~f, Y~g, and
P(X=0.4)=P(Y=0.4)=1. Then, X and Y are independent, their
medians are <0.5, but there sum has a median >0.5.

Its different for "multiplicative", since in general from
X~f, Y~f, P(X>=0.5)<=0.5, and P(Y>=0.5)<=0.5 it follows
that P(XY>=0.5) <= P(X>=p or Y>=q) if p*q=0.5. Thus, if there
are numbers p and q with that property, such that
P(X>=p) + P(Y>=q) <= 0.5, then the median of XY will be <=0.5.

You might argue that there is a relationship between additive
and multiplicative scale through a log-transformation (note that
the median is stable under monotone transformations). However,
I assume there is no obvious formulation of the above statement

There is no way of carrying convolutions out in R directly; you'd
need to do numerical integration to do that, e.g. using
integrate().

HTH

Thomas

---

Thomas Hotz
Research Associate in Medical Statistics
University of Leicester
United Kingdom

Department of Epidemiology and Public Health
Leicester
LE1 6TP
Tel +44 116 252-5410
Fax +44 116 252-5423

Division of Medicine for the Elderly
Department of Medicine
The Glenfield Hospital
Leicester
LE3 9QP
Tel +44 116 256-3643
Fax +44 116 232-2976

> -----Original Message-----
> From: Salvatore Barbaro [mailto:sbarbar at gwdg.de]
> Sent: 24 July 2003 12:56
> To: r-help at stat.math.ethz.ch
> Subject: [R] median and joint distribution
>
>
> Dear R-"helpers"!
>
> May I kindly ask the pure statistics-experts to help me for a
> purpose which first part is not directly concerned with R.
> Consider two distribution functions, say f and g. For both, the
> median is smaller than a half. Now, the multiplicative or additive
> say h, whereas the median of h is greater than a half. Does
> anybody know under which circumstances such a construction of h is
> possible (my intuition is that it depends on the correlation of f
> does anybody know whether or how such a construction can be done
> with R. Thanks in advance.
>
> s.
>
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> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
>

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