} be a sequence of independent 
with its probability 
. Claim that 
in probability but 
in probability. (15%)| 所組別： | 數學學系 | 科目： | 機率 | 考試時間： | 6月14日第1節 |
| 1. | Describe the Borel - Cantelli Lemma and prove it. (15%) |
| 2. | Let {} be a sequence of independent with its probability . Claim that in probability but in probability. (15%) |
| 3. | Let  be
arbitrary events with  and  where  is the complement of . Show that  (15%) |
| 4. | Let 
be  with  where  claim that  as  . (10%) |
| 5. | If  in
distribution and 
in distribution then  in distribution. (15%) |
| 6. | Suppose that  is a sequence of constants tending to  , b is a fixed number, and  in distribution. Let g
be a function of a real variable which is differentiable and whose derivative  is continuous at b.
Show that  in
distribution. (15%) |
| 7. | Let  be
a random variable and let g be a nonnegative Borel-measurable function on  . |
| (a) | Show that if g is even  and nondecreasing on  then for every  ,  . |
| (b) | Use the inequality in to prove that  in probability if  as  . (15%) |
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