Probability of n adults in household I have a bunch of households, and I pick one at random. Each household has 1-9 people in it, and each person can either be an adult or a child. The probability of picking a household of size $k$ is known, and is $p_k$ (with $\sum_{k=1}^9 p_k = 1$), and the proportion of people across all households that are adults is also known, and is $q$. I don't have more information about the joint distributions.
What I would like to simulate is either:
1) The number of adults in a house, given you're in a household of size $k$
2) The number of adults in a house given you're in a household with $k$ children
The two might be equivalent - I'm not entirely sure.
Do I have enough information here to do this/ something close to this? Are there any assumptions I could make that would make this doable (and I can then decide whether those assumptions are feasible)?
Edit: I may also have access to the proportion of households with 0,1 or 2 adults. If I assumed every household had at most 2 adults (not true, but I'll go with it for now), is there anything I could do with that?
 A: Let $s_j$ be the expected proportion of adults in households of size $j$. Then if we let $q$ be the fraction of the population that are adults, the following holds:
$$q = \frac{\sum_{j=1}^9p_j*s_j*j}{\sum_{j=1}^9p_j*j}$$
Your goal is to simulate $X_k$, a random variable representing the number of adults in a house of size $k$. Two problems present themselves:


*

*$s_1$, ..., $s_9$ are unknown.

*Even if we could estimate the $s_j$, the distribution of $X_k$ conditional on $s_k$ is unknown.


However, not all hope is lost. Let's start with problem 1. We can simplify problem 1 a little bit. For example, we can probably safely assume $s_1=1$. Because of our above equation for $q$, we need only solve for 7 of the remaining 8 $s_j$s. You can think about layering reasonable assumptions onto this. For example, if you wanted to assume that every household of size 3 or more had $s_j=2/j$, you could then solve for $s_2$. This assumption excludes the possibility of single-parent households when there are 2 or more children, so I wouldn't advise making this assumption, but I wanted to give one example of an assumption that makes this solvable. You could instead assume $s_j=(2-\epsilon)/j$ for $j>2$, and this would perhaps be a better approximation and would reduce the number of unknown parameters to only 2 ($s_2$ and $\epsilon$). One more piece of information or assumption would be sufficient to solve the puzzle at this point.
As for problem 2, even if we are able to obtain $s_1$,...,$s_9$, the distribution of $X_j$ is still unknown. Sure, if we assumed $X_j\sim \mbox{Binomial}(j,s_j)$, then you could simulate $X_j$, but that is likely not a safe assumption as this could potentially place significant probability on having 3 or more adults in the same household, especially for large $j$. This is where the assumption about having no households with 3+ adults would come into play. This would again simplify the problem, but more information would be needed to fully specify the distribution even if we restricted the support of $X_j$ to be $(1,2)$.
So in short, you do need more information to solve this problem. However, I hope that the above framework can help you think about how the assumptions you make could easily turn this into a solvable problem.
