# A basic question on Poisson random variable

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

## 1 Answer

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.

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