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