Lad $X_1,...,X_n$ be i.i.d. with the distribution $\mathcal{Poiss}(\lambda)$, for $\lambda>0$.
- Write the joint pdf $(X_1,...,X_n)$ and argue that for $$P((X_1,..,X_n) \in \left \{ 0,1,2,..., \right \}^n)=1$$
for a fixed $(x_1,...,x_n) \in \left \{ 0,1,2,..., \right \}^n$
Concerns
I am unsure on what I am even asked to show. I get the Poisson distribution definition and the support of it. But I am not sure on what is asked to show to even begin to show it is indeed equal to $1$. I can probably do the calculations and manipulation when I get it in another form.
The probability mass function of a vector of random variables will equal the joint probability mass function of the random variables.
The joint probability mass function of independent random variables shall be the product of the probability mass functions of those random variables.
The probability mass functions of identically distributed random variables are identical, and these are a Poisson distributed random variables with parameter $\lambda$.
So....
$$\begin{align}\mathsf P(\langle X_1,\ldots,X_n\rangle{=}\langle x_1,\ldots, x_n\rangle) ~&=~ \mathsf P\left(\bigcap\limits_{i=1}^n( X_i{=}x_i)\right)\\&=~ \prod_{i=1}^n \mathsf P(X_i{=}x_i)\\[0ex]&~~\vdots\end{align}$$