In each case below, derive the probability limit and an asymptotic distribution for \(n^{-1} \sum_{i=1}^{n} X_{i}\) and a limiting distribution for the random variable \(Y_{n}\), if they can be defined.

(a) \(X_{i}^{\prime}\) s iid Bernoulli \((p), Y_{n}=\frac{n^{-1} \sum_{i=1}^{n} X_{i}-p}{n^{-1 / 2}(p(1-p))^{1 / 2}}\)

(b) \(X_{i}^{\prime}\) s iid \(\operatorname{Gamma}(\alpha, \beta), Y_{n}=\frac{n^{-1} \sum_{i=1}^{n} X_{i}-\alpha \beta}{n^{-1 / 2} \alpha^{1 / 2} \beta}\)

(c) \(X_{i}^{\prime}\) s iid Uniform \(

(a, b), Y_{n} \frac{n^{-1} \sum_{i=1}^{n} X_{i}-.5(a+b)}{(12 n)^{-1 / 2}(b-a)}\)

(d) \(X_{i}^{\prime}\) s iid Geometric \((p), Y_{n}=\frac{n^{-1} \sum_{i=1}^{n} X_{i}-p^{-1}}{\left(n p^{2}ight)^{-1 / 2}(1-p)^{1 / 2}}\)