I am watching a statistics lecture, and the professor has provided an example of data he collected of number of siblings (including ones self) people have. He asserts that directly estimating the PMF is quite difficult especially if the sample size is small because of the many parameters that would need to be estimated. I understand this part. He then suggests that one use a Poisson distribution to model the distribution we are concerned with. I am confused here.
My prior knowledge leads me to believe that Poisson distributions are best at modeling discrete events that occur in a fixed interval of time. I don't see how that would be accurate modeling the number of siblings one has, since there is no "interval of time" involved. Wouldn't a Gaussian distribution be much more suitable for the task?
Gaussian random variables are continuous and two sided, where here by assumption we're always going to have that the number of siblings is greater than or equal to 1. Sampling from a Gaussian will also almost always return non-integer values, which may not be good for what you're trying to estimate. Equally, the Gaussian is symmetric, where here values will never be below 1, and the curve has no reason to be symmetric about its mean.
On the other hand, a Poisson will immediately give you a natural number which is appropriate for the number of siblings. Note also that this is particularly important since the number of common outcomes is very small - almost all families will have fewer than 10 children, so a Gaussian will be providing much more than you need.
You could view the interval as the time that the parents are trying to have children, so the time the family is still growing.