> Yes, birth dates of themselves and family members, which are randomly distributed in the population

Nope - not uniformally random, there's literally a non uniform age distribution, a specific set of months, a capped number of days, and various "peak" birth seasons.

    >> > those commonly picked sets of numbers
    > What are those numbers?

One way to find out is to get a math degree and moonlight as a consultant for, say, the local uni math department and perform quasi regular spot checks on the state lottery games.

There are many lottery variations and differing games show interesting patterns of choice - although more and more people these days simply get machine generated tickets .. this expands the surface of interest as not only the game results themselves need to be checked for bias but also the various networked game terminals for their random generation and for local tampering.

It's probable that anyone who has done such work is bound by NDA's and agreements not to dump mass data aquired through work.