Eh, thats not very good evidence of the claim, and wouldn't over 50% be first born anyway? I dont know the average distribution of child-sizes but I would assume almost half of people in the US are first borns (including only-children).
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
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.
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
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.