As I was looking to understand the concept of "family of distributions", I stumbled upon this answer. However, I was a bit confused with answer and I'm hoping that someone may be able to clarify for me if my understanding is wrong or close with as little math as possible.
In pseudo-layman's term, my understanding of "family of distributions" is as follows:
Suppose that I have Y ~ Chi-Squared(n). We know that the relationship between the Chi-Squared Distribution and the Gamma Distribution is that Chi-Squared(n) = Gamma(n/2, 1/2).
So my question is: "Is the Chi-Squared Distribution in the same family as the Gamma Distribution in the sense the Chi-Squared Distribution IS a Gamma Distribution when the Gamma Distribution holds certain parameters?
If this is correct, I can also say that the Erlang Distribution is also in the same family as the Chi-Squared Distribution and the Gamma Distribution because the Erlang Distribution is just a Gamma Distribution where the alpha parameter of the Gamma Distribution [Gamma(alpha, beta)] is an integer.
Therefore, my understanding at the moment is that a family of distributions is really just a set of distributions that come from a more general distribution when that general distribution holds certain parameters (e.g Gamma Distribution being the general distribution and one can derive the Chi-Squared and Erlang when they put certain parameters in for Gamma).
