(d) (10%) Does Y_n converge in distribution? If yes, what distribution does it converge to? Prove the convergence or divergence.

2. (50%) Let $X1, X2, \cdots, X_n$ be a random sample from a normal distribution $$f(x; \theta) = (2\pi\theta)^{-1/2} \exp \left( -\frac{1}{2\theta} (x - \mu)^2 \right), x \in R; \mu \in R, \theta > 0.$$ Let the distribution function of $f(x; \theta)$ be denoted as $\Phi(x; \mu, \theta)$. (a) (10%) Find the maximum likelihood estimators $\hat{\mu},\hat{\theta}$ of $\mu ,\theta$

(d) (10%) Does \( Y_{n} \) converge in distribution? If yes, what distribution does it converge to? Prove the convergence or divergence.

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