22. In each case below, derive the probability limit and an asymptotic distribution for n1 Pn i1...

22. In each case below, derive the probability limit and an asymptotic distribution for n1 Pn i¼1 Xi and a limiting distribution for the random variable Yn; if they can be defined.

(a) Xi 0

s iid BernoulliðpÞ; Yn ¼

n1 Pn i¼1 Xi  p n1=2ð Þ pð Þ 1  p 1=2

(b) Xi 0

s iid Gammað Þ a; b ;Yn ¼

n1 Pn i¼1 Xi  ab n1=2a1=2b

(c) Xi 0

s iid Uniformð Þ a; b ;Yn n1 Pn i¼1 Xi  :5ð Þ a þ b

ð Þ 12n 1=2

ð Þ b  a

(d) Xi 0

s iid GeometricðpÞ;Yn ¼

n1 Pn i¼1 Xi  p1 np2 ð Þ1=2

ð Þ 1  p 1=2