# properties of Poisson random variables

###### Proposition 1.

If $X_{1},X_{2}$ are independent Poisson random variables with parameters $\lambda_{1},\lambda_{2}$, then $X_{1}+X_{2}$ is a Poisson random variable with parameter $\lambda_{1}+\lambda_{2}$.

###### Proof.

Let $X:=X_{1}+X_{2}$ and $\lambda:=\lambda_{1}+\lambda_{2}$, let us calculate the distribution function of $X$:

 $\displaystyle F_{X}(x)$ $\displaystyle=$ $\displaystyle P(X\leq x)=P(X_{1}+X_{2}\leq x)=\sum_{i=0}^{x}P(X_{1}+X_{2}=i)$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{x}\sum_{j=0}^{i}P(X_{1}=j\mbox{ and }X_{2}=i-j)=\sum_% {i=0}^{x}\sum_{j=0}^{i}P(X_{1}=j)P(X_{2}=i-j)$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{x}\sum_{j=0}^{i}\frac{e^{-\lambda_{1}}\lambda_{1}^{j}% }{j!}\frac{e^{-\lambda_{2}}\lambda_{2}^{i-j}}{(i-j)!}=\sum_{i=0}^{x}\sum_{j=0}% ^{i}\frac{e^{-\lambda}}{i!}\binom{i}{j}\lambda_{1}^{j}\lambda_{2}^{i-j}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{x}\frac{e^{-\lambda}}{i!}\sum_{j=0}^{i}\binom{i}{j}% \lambda_{1}^{j}\lambda_{2}^{i-j}=\sum_{i=0}^{x}\frac{e^{-\lambda}}{i!}(\lambda% _{1}+\lambda_{2})^{i}=\sum_{i=0}^{x}\frac{e^{-\lambda}}{i!}\lambda^{i}.$

As a result, $X$ is a Poisson random variable with parameter $\lambda$. Notice that in the fifth equation, we used the assumption that $X_{1}$ and $X_{2}$ are independent. ∎

As a corollary, any sum of independent Poisson random variables is Poisson, with parameter the sum of the parameters from the independent random variables.

###### Proposition 2.

A Poisson random variable is infinitely divisible.

###### Proof.

Let $X$ be a Poisson random variable with parameter $\lambda$. Let $n$ be any positive integer. Let $X_{1},\ldots,X_{n}$ be independent identically distributed Poisson random variables with parameter $\frac{\lambda}{n}$. Then the sum of these random variables is easily seen to be Poisson, with parameter $\lambda$, and is therefore identically distributed as $X$. ∎

Title properties of Poisson random variables PropertiesOfPoissonRandomVariables 2013-03-22 18:50:55 2013-03-22 18:50:55 CWoo (3771) CWoo (3771) 4 CWoo (3771) Derivation msc 62E15