= Solution
Let $m=1/\sigma^2$. Within block $i$, the $m$ replicated observations have dispersion $a_i$ and common natural parameter $\theta_i$. Evaluating their joint MGF at the blockwise argument $\sigma^2t_i$ gives
$$
\left[
\exp\!\left\{
\frac{b(\theta_i+a_i\sigma^2t_i)-b(\theta_i)}{a_i}
\right\}
\right]^m
=\exp\!\left\{
\frac{b(\theta_i+a_i\sigma^2t_i)-b(\theta_i)}{a_i\sigma^2}
\right\}.
$$
Multiplying over blocks reproduces the MGF of $Y$ from part (a). Since moment-generating functions uniquely determine distributions in this family, $\bar Y\overset d=Y$.
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