臺灣大學數學系
八十八學年度第二學期碩博士班資格考試試題 統計與機率
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機率(機率組)
(25/105) For any random variable on denote the characteristic function of
by .
(1.1)
Show that is uniformly continuous for . (1.2)
Let be a Normal random variable with mean μ and variance . Find . (1.3)
Apply (1.2) to evaluate , where is a Normal random variable with mean and variance . State clearly which properties or facts you are using to solve this question.
1.
(30/105) Let be a fixed probability space.
(2.1)
Let . Prove Borel-Cantelli lemma: If , then
. (2.2)
Let be a sequence of random variables. Apply (2.1) to prove that as
converges to in probability, denoted as , if and only if each subsequence contains a further subsequence which
converges to almost surely.
(2.3)
Let be two sequences of random variables such that
. Show that and .
(Note that if you cannot prove (2.2), you can still apply it to establish (2.3).) 2.
(20/105) Let be the product of Bernoulli measures of on such that
3.
. Put .
(3.1)
Denote by the conditional distribution of given . Find .
(3.2)
Evaluate by using the limiting behavior of .
機率(統計組)
(15/105) (1.1) Let be a Normal random variable with mean μ and variance . Find its
moment generating function .
(1.2) Apply (1.1) to evaluate , where is a Normal random variable with mean and variance .
1.
(15/105) Let be the product of Bernoulli measures of on such that
. Put .
(2.1) Denote by the conditional distribution of given . Find .
(2.2) Evaluate .
2.
統計(機率組做 1,2 題, 統計組全做)
(15 points) Assume are i.i.d. according to , .
(i)
(5 points) Find the maximum likelihood estimator of .
(ii)
(10 points) It is known that the limit distribution of is exponential distributed with parameter θ (i.e., The density function .).
Here is the largest order statistic. Use this result to determine the
nondegenerate limit distribution of under proper normalization.
1.
(15 points) Let and be independent normal and , respectively, and consider the test of against with rejection region
where , , ,
, and where is a standard normal random variable. Show that this test has asymptotic level α.
2.
(10 points) The random variable has a binomial distribution with an unknown number θ of trials, and known probability of success, . (Namely, .) Find an approximated confidence interval of θ of the form where is to be
determined.
3.
(15 points) Suppose that the independent pairs of random variables
are such that and are independent in and , respectively.
(i) (8 points) Use the method of moment to derive the estimate β.
(ii) (7 points) Is the estimate obtained in (i) consistent?
4.
(15 points) In life-testing experiments, it is quite often that the experiment is terminated whenever the first failures have occurred among tested units. Suppose the survival time of a particular units follows an exponential distribution . (i.e.,
.)
(i) Derive the maximum likelihood estimate of θ under the above setting.
(ii) Discuss whether the resulting estimator is consistent when . 5.
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