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Comment by swwu | Hacker News Reader

The "average number of own children under 18 per family" for "familes with own children under 18" was 1.86 in 2009 (and has been within 0.1 of that since 1978), which means about 1/1.86=~54% of all children are first-borns...

I don't think we have enough information here to actually verify but I suspect 80% vs 54% is still a statistically significant difference.

But it's pretty obvious that the 80% number is an arbitrary guess, so I'm not sure exactly where I stand here.

source: http://www.census.gov/population/socdemo/hh-fam/fm3.csv

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No, it depends on the distribution. Let's say (as a first guess) that number of children follow a poisson distribution. A poisson distribution with mean of 1.84 has 74% of children being first borns.

There are a lot of reasons why a poisson model probably isn't great, but it's worth pointing out that a model with more density near zero is probably likely and implies that first borns are more popular than the they look.

74% versus 80% isn't anything to write home about.

Moreover, as Mz points out, only-children are a totally different case.

A Poisson distribution with a mean of 1.84 results in about 45% of children being first born. In fact, the best you can do with any distribution is 54%. For example, if there are 10k couples, then there will be 18.4k children born, at most 10k of which can be first born; and 10/18.4 = 54%.

In fact, the only thing that matters for the distribution (once you know the mean) is the number of childless families, since they could have had a first born but didn't.

I haven't been able to figure out where 74% comes from. Is there some way to sample on a per-family basis to come up with this?

edit: spelling

More like a fundamentally stupid mistake of inverting my parameters when I checked the number, getting a result in tune with the article, and then not thinking it through at all. It'd be possible to have a whole great deal of families with 0.1st children, but that's just the egg on my face.

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.

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