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Let be a sequence of random variables satisfying Show that if 3

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臺灣大學數學系

八十七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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Probability

Let be a random sample from logistic distribution with cdf

. Let .

(11) Show .

(12) Show converge to a limiting distribution.

(13) Find .

1.

Let be a sequence of positive, integral-valued random variables such that

as , where . Let be a sequence of

independent, identically distributed random variables with and , . Find the asymptotic distribution of as . Justify your answer.

2.

Let be a sequence of random variables satisfying

Show that if 3.

Suppose that and are independent random variables with a common distribution function that is positive and continuous. What is the conditional probability of

given the random variable ?

4.

Statistics

Let be a random sample from a population with density

, , .

Show that is a method of moment estimate of θ.

1.

Show that 2.

1.

(2)

in law of large number.

Let , , , where the are independent

variables, . Derive the likelihood ratio test of versus for some , is used.

2.

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