The poisson, exponential, and gamma distributions can be derived from the binomial distribution with assumptions.
The chi-squared distribution is a special case of the gamma distribution. The chi-squared distribution is also the sum of squared standard normal random variables. However, the normal distribution can be derived from the binomial distribution too.
So the rough relationship between these members of the exponential family of distributions looks like this to me, both ending up at chi-squared distributions:
(1) Binomial --> poisson --> exponential --> gamma --> chi-squared
(2) Binomial --> normal -(squaring)-> chi-squared
Squaring a distribution seems like a big deal, yet one can start with binomial and end up at chi-squared either through squaring or not. This seems very counterintuitive - I thought this would be more likely:
(1) Binomial --> poisson --> exponential --> gamma --> chi-squared
(2) Binomial --> normal -(squaring)-> NOT chi-squared
OR
(1) Binomial --> poisson --> exponential --> gamma --> NOT chi-squared
(2) Binomial --> normal -(squaring)-> chi-squared
Even more confusingly, the chi-squared can under certain conditions approximate the normal distribution (yet the standard normal variables were squared!)
I would be very grateful if someone could help me understand what I'm missing here! Thanks!