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Given a Poisson random variable $X$ with parameter $\lambda$, I want to write $X=X_1+X_2$ where $X_i$s are independent Poisson random variables with parameter $\lambda_i$s such that $\lambda=\lambda_1 + \lambda_2$

I understand this intuitively. But given only a function from $\Omega \to \Bbb R$ how to write these two functions

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Consider an i.i.d. sequence of Bernoulli random variables $(U_i)_{i\geqslant1}$ with parameter $\lambda_1/(\lambda_1+\lambda_2)$, independent of $X$, and define $$ X_1=\sum_{i=1}^XU_i,\qquad X_2=\sum_{i=1}^X(1-U_i). $$ Then $(X_1,X_2)$ is as desired.

This operation is called thinning and is explained in every decent book on Poisson processes. The fact that $X_1$ and $X_2$ have the desired distributions and are independent is a good, elementary, exercise on discrete random variables.

Naturally, this construction assumes that the probability space $(\Omega,\mathcal F,P)$ is "large enough" so that it is possible to define the needed sequence $(U_i)_{i\geqslant1}$ on it. Anyhow, if the space is "too small" (say $\Omega=\mathbb N_0$ with the discrete sigma-algebra and $X:\Omega\to\mathbb R$ the identity), it may be impossible to decompose $X$ as desired.

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  • $\begingroup$ yes I understood after posting this $\endgroup$ – RIchard Williams Sep 27 '13 at 13:41
  • $\begingroup$ And so what? $ $ $\endgroup$ – Did Sep 28 '13 at 9:08

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