Question

For each proposed function b shown below, indicate (based only on its second derivative) whether it could potentially be a log-partition function of some exponential family with dispersion \phi = 1.

b(\theta )=\theta ^2-2\theta +1



Valid

Invalid
unanswered
b(\theta )=\sqrt{\theta }



Valid

Invalid
unanswered
b(\theta )=\ln \theta



Valid

Invalid
unanswered
b(\theta )=\theta



Valid

Invalid

Answer 1

Invalid