Re: Compound Distributions?

Carsten Steinhoff <carsten.steinhoff@xxxxxxxxxxxxxxxxxxxxxx> wrote:
>
>Hello,
>
>I want to fit some distributions on a dataset. My tests up to now
>resulted in something like:
>
>Distribution A fits very well from 0 to K
>Distribution B fits very well above K
>
>In a second step I have to draw random numbers from the chosen
>distribution (drawing uniform distr. rn and put them into the F^-1).
>
>Now the Problem is ... : I want to put distr A and B together.
>
>Its not sure that K is identical to the point of intersection! So it
>might be that the combination of A and B is not stetic in K.
>
>So how to draw my random numbers??? If I put e.g a 0.88 into the
>"common" inverse in some cases not necessarily a value results - in
>other cases there could result two (depending whether the intersection
>is left or right from K).
>
>I think there should be something to google about, but when I search for
>"compound distributions" I find what I would call "mixing" (e.g. F = F_A
>+ F_B)
>
>Who can help?
>
>Thanks a lot!!!


Let p_A and p_B be the probability densities you are talking about,
and let
I_A = integral(p_A(x), x=0..K),
I_B = integral(p_B(x), x=K..upper_border)

The density you want to generate is

p_A(x)/(I_A+c*I_B) if x in [0,K]
c*p_B(x)/(I_A+c*I_B) if x in [K,upper_border]

for some positive number c.
Right?

Generate a rn.
If rn < I_A/(I_A+c*I_B) then
generate a new rn to construct x in [0,K] following p_A
else
generate a new rn to construct x in [K,upper_border] following p_B
endif

.

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