I am doing original research for my undergrad capstone course, and I was wondering if anyone here could help me see where to go next based on what I already know. I am trying to identify a distribution, and the most useful information I know about it is. X(t) is a random variable $$ E(X|at) = \sqrt a E(X|t) $$ $$ V(X|at) = aV(X|t) $$ And using those two you can show $$ E(X^2|at) = aE(X^2|t) $$ After messing around with this for awhile, I found it interesting what distributions these properties hold for and which distributions they don't. They hold for exponential, gamma, inverse gamma, and Fréchet, which all have similar PDFs. but they also hold for generalized Pareto, log-logistic, and Lomax to name a few. So finally, my question, what can I deduce about the distribution given the information above? Thanks.
(also, from looking at the numerical analysis I've done, it looks like the distribution might be inverse gamma, but I'm not very close to proving that yet)