| 91年 6月19日 08:30~10:00 應用數學系 (選考) | 誠實是我們珍視的美德， 我們喜愛「拒絕作弊，堅守正直」的你！ |
| 科目： 機 率 |
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1.
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Let X1 , X 2 , … be random variables. Claim that (15%) | |
| (a) |
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| (b) | lim n sup X n is also a random variable | |
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2.
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Describe the Fatou's Lemma and then prove it. (15%) | |
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3.
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In a sequence of Bernoulli trials with common success probability p, let the random variable X be defined as X = n if the first occurrence of the sequence SF (Success, Failure) occurs at trials n-1 and n. Derive the probability generating function E ( t x ) of X. (15%) | |
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4.
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Prove that 
if and only if  and  .
(15%) |
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5.
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Let X1 , … , Xn
, … be iid from 
. Describe the asymptotic normality of the maximum likelihood estimator
(MLE) of  and then prove it.
(20%) |
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6.
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Let X , X 1 , X 2 , … be random variables and let c be a constant. (20%) | |
| (a) | Claim that Xn → c in distribution implies Xn → c in probability. | |
| (b) | Give a counterexample about the statement that Xn → X in distribution implies Xn → X in probability. | |
is a random variable for
all 
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