# Independent vs. Dependent Compound Poisson Distributions

I have an issue with a section of some Actuarial lecture notes that I am reading. Here are the snippets from the notes:

"Consider a portfolio consisting of $n$ independent policies. The aggregate claims from the $i$-th policy are denoted by the random variable $S_{i}$, where $S_{i}$ has a compound Poisson distribution with parameters $\lambda_{i}$ (all i.i.d), $\textbf{not known}$, and the CDF of the individual claim amounts distribution is $F(x)$, known."

The notes then go on to say that this implies that all of the $S_{i}$s are i.i.d. This makes intuitive sense.

However, the next section has the same set-up, but now the Poisson distribution parameters are all $\lambda$. The notes state "If the value of $\lambda$ were known, then the $S_{i}$ are i.i.d". I.e. the $S_{i}|\lambda$ are i.i.d. Implying that the $S_{i}$ themselves (i.e. with $\lambda$ $\textbf{not known}$) are dependent. This seems to contradict the first section.

If you can help me get my head around this, intuitively, that would be a great help.

Thanks very much.

In the second case, with $\lambda$ not known, the $S_i$ are dependent because observing one $S_i$ tells you something about $\lambda$ which in turn tells you something about the other $S_i$'s. If $\lambda$ is known, or each has a separate $\lambda_i$, then this flow of information is broken and they are independent.