Isn't Poisson rate-based, ie events over a fixed time period? I'm having trouble seeing where you got the rate from (children per lifetime? but that's hardly a "fixed time period") - all we have is a single number "children/household" with no time dimension.
And most other distributions, ie other than the uniform I'm assuming and the Poisson you're using, require >1 parameters. Since we only have one piece of data (the mean) we would have to make assumptions with these distributions that make no more sense than simply assuming a uniform distribution to begin with.
You'll have to excuse me for any incredibly stupid intuitive leaps; I haven't had a chance to wake up fully yet.
And most other distributions, ie other than the uniform I'm assuming and the Poisson you're using, require >1 parameters. Since we only have one piece of data (the mean) we would have to make assumptions with these distributions that make no more sense than simply assuming a uniform distribution to begin with.
You'll have to excuse me for any incredibly stupid intuitive leaps; I haven't had a chance to wake up fully yet.