I’ve this drawback, of which I do know the answer, however I am searching for the mathematically correct approach of writing it.

Say I’ve a (infinite) inhabitants of individuals, the place every particular person is labeled by two unbiased variables: $𝑁$ (a constructive integer) and $okay$ (steady). Additionally, every particular person has a sure amount of cash, which is a operate of $N$ solely, say $m=g(N)$.

Now in some unspecified time in the future in my derivations I’ve to make use of the amount $Psi(N,okay)$, which is the distribution of the fraction of the entire cash within the inhabitants. Meaning $Psi(N,okay)dNdk$ represents the factor of fraction of the entire cash possessed by folks of worth $N$ and $okay$. (I deal with $N$ as a steady variable for simplicity.)

So I can develop $Psi(N,okay)$ within the following approach

$Psi(N,okay)=frac{g(N)f_{Nk}(n,okay)}{int g(N)f_{Nk}(n,okay)dN dk }=frac{g(N)f_N(okay)f_N(N)}{int g(N)f_N(okay)f_N(N)dN dk } = left (frac{g(N)f_N(N)}{int g(N)f_N(N)dN } proper ) f_N(okay)=F(N)f_N(okay)$

with $f_{…}$ the likelihood distributions.

The time period in brackets, that I outline as $F(N)$, represents the fraction cash possessed by folks of worth $N$. That is one thing that I do know (as an instance it is a prior information).

Now that is my drawback: $N$ by itself truly would not have a likelihood distribution (it may possibly take any worth between 1 and infinity and the variety of folks having worth $N$ will increase with $N$). And but $F(N)$ is outlined and finite.

So, would anybody know a rigorous strategy to write $Psi(N,okay)=F(N)f_N(okay)$, with out having to invoke $f_N$ at any step of the derivation? Or every other good approach? It might sound intuitive, however not trivial sufficient to not present it rigorously…

Many thanks!