Consider a random variable Y with distribution p_\theta (y) for some \theta, coming from a canonical exponential family.

Question

Let Z=Y+a, where a is a constant. Denote by q_\theta (z) the density of Z,\, which is parametrized by \theta.

Is q_\theta also a member of some canonical exponential family?

Yes

No
unanswered
Now instead suppose Z=\lambda Y, where \lambda \neq 0 is constant. This again determines some density \widetilde{q}_\theta (z) of Z.

Is \widetilde{q}_\theta also a member of some canonical exponential family?

Yes

No

Answer 1

Yes