[R] Generate random numbers up to one
(Ted Harding)
ted.harding at nessie.mcc.ac.uk
Tue Mar 6 23:01:39 CET 2007
On 06-Mar-07 Alberto Monteiro wrote:
> Ted Harding wrote:
>>
>> And, specifically (to take just 2 RVs X and Y), while U = X/(X+Y)
>> and V = Y/(A+Y) are two RVs which summ to 1, the distribution of U
>> is not the same as the distribution of X conditional on (X+Y = 1).
>>
> This question
Which question? There are (implicitly) two questions there!
> appeared in October 2006, and the answer
To the second question (X conditional on X+Y=1)
> was the Dirichlet distribution with parameters (1,1,1...1):
>
> http://en.wikipedia.org/wiki/Dirichlet_distribution
>
> It's the distribution of uniform U1, U2, ... Un with the
> restriction that U1 + U2 + ... + Un = 1.
Indeed, and the resulting (U1,U2,...,Un) is uniformly distributed
on the simplex U1+U2+...+Un=1. For n>2, however, the resulting
marginal distribution of (say) U1 conditional on (U1+U2+...+Un=1)
is no longer uniform (that only holds for n=2, as in my example).
For n=3 this is easy to see: P[U1 > u1] is the area of the triangular
simplex between its vertex at (1,0,0) and the line from (u1,1-u1,0)
to (u1,0,0), and this is equal to (1 - u1)^2, so the density of U1
is f(u1) = 2*(1-u1). In general, the marginal density of U1
in the n-dimensional Dirichlet is (n-1)*(1-u1)^(n-2).
But the aim was to illustrate Petr Klasterecky's point that
"sum(x) is a random variable as well and dividing by
sum(x) does not preserve the original distribution
data were generated from."
namely to show two ways of generating RVs distributed on
U1 + U2 + ... + Un = 1, starting from independent RVs, which
result on two different distributions, and to give an example
where dividing by sum(x) can be seen to "not preserve" the
distribution.
Indeed, I think there is sometimes a confusion between this
question and the really unrelated question: Given non-negative
numbers V1, V2, ..., Vn, how can we convert then to a probability
distribution? To which the answer is, of course, divide by their
sum.
With best wishes,
Ted.
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Date: 06-Mar-07 Time: 22:01:34
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