# [R] Looking for simpler solution to probabilistic question

G. Jay Kerns gkerns at ysu.edu
Tue Jan 15 22:31:47 CET 2008

```Dear Rainer,

If X1 is random on 1:20 with probabilities proportional to
dnorm(,10,2), and X2 is random on 10:30 with probs proportional to
dnorm(,15,1), and the object is to find out the probability
distribution of X1 + X2, then a very quick way is with package distr:

library(distr)
X1 <- DiscreteDistribution(1:20, prob = dnorm(1:20,10,2))
X2 <- DiscreteDistribution(10:30, prob = dnorm(10:30,15,1))
X <- X1 + X2

plot(X) # plot the probabilites

d(X)(11:50) # look at probs individually

Notice that the answer from distr is different from the answers posted
previously.  The answers will be the same if the line

p <- matrix(p1, ncol=length(p1), nrow=length(p2), byrow=TRUE) + p2

is replaced with

p <- matrix(p1, ncol=length(p1), nrow=length(p2), byrow=TRUE) * p2

(the only difference is the multiplication by p2 instead of addition
with p2).  This is following from an assumption of independence
between X1 and X2. Otherwise, you would seem to need to know the joint
distribution of X1 and X2, which isn't mentioned in your problem
description.  Is there additional information about the problem that
would suggest adding p2 rather than multiplying?

I hope this helps,
Jay

On Jan 15, 2008 8:33 AM, Rainer M Krug <R.M.Krug at gmail.com> wrote:
> Berwin A Turlach wrote:
> > G'day Rainer,
> >
> > On Tue, 15 Jan 2008 14:24:08 +0200
> > Rainer M Krug <R.M.Krug at gmail.com> wrote:
> >
>
> >
> >> ager <- range(age1) + range(age2)
> >> ager <- ager[1]:ager[2]
> >> pp1 <- c(cumsum(p1), rev(cumsum(rev(p1))))
> >> pp2 <- c(cumsum(p2[-21]), rev(cumsum(rev(p2)))[-1])
> >> pr <- pp1+pp2
> >> pr <- pr/sum(pr)
> >
> >> all.equal(p, pr)
> > [1] TRUE
> >> all.equal(age, ager)
> > [1] TRUE
> >
> >
> > If this is more elegant is probably in the eye of the beholder, but it
> > should definitely use less memory. :)
>
> Thanks - interesting approach which is different to using the outer()
>
> >
> > BTW, I am intrigued, in which situation does this problem arise?  The
> > time it takes the second process to finish seems to depend in a curious
> > way on the time it took the first process to complete.....
>
> These are two growth processes, where the first one is seedling growth
> up to a certain X and the second one is adult growth from size X onwards
> until it reaches a given final size.
>
> I hope my calculations fit the process...
>
>
>
> >
> > Cheers,
> >
> >       Berwin
> >
> > =========================== Full address =============================
> > Berwin A Turlach                            Tel.: +65 6516 4416 (secr)
> > Dept of Statistics and Applied Probability        +65 6516 6650 (self)
> > Faculty of Science                          FAX : +65 6872 3919
> > National University of Singapore
> > 6 Science Drive 2, Blk S16, Level 7          e-mail: statba at nus.edu.sg
> > Singapore 117546                    http://www.stat.nus.edu.sg/~statba
>
> ______________________________________________
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> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

--

***************************************************
G. Jay Kerns, Ph.D.
Assistant Professor / Statistics Coordinator
Department of Mathematics & Statistics
Youngstown State University
Youngstown, OH 44555-0002 USA
Office: 1035 Cushwa Hall
Phone: (330) 941-3310 Office (voice mail)
-3302 Department
-3170 FAX
E-mail: gkerns at ysu.edu
http://www.cc.ysu.edu/~gjkerns/

```