八十八學年度第一學期碩博士班資格考試試題 統計與機率

臺灣大學數學系

八十八學年度第一學期碩博士班資格考試試題 統計與機率

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*機率

(i) Let be a probability space and . Prove

that is a sub-σ-field of .

(ii) Let be a sequence of independent random variables defined on and be a tail event of . Show that or 1.

1.

Let be a sequence of random variables converges to almost surely. Show that converges to in probability. The converse is true or false?

2.

Let be a sequence of indepent random variables. Show that converges

almost surely if and only if converges in probability.

3.

Let be a sequence of i.i.d. random variables with and

, . Let . Show that

converges in distribution to .

*統計 4.

Let be a random sample from

(i) Find the maximum likelihood estimator for the parameter θ.

(ii) Show that is a biased estimator.

5.

Let denote a random sample from the distribution that has

Find the best critical region for testing against assume 6.

.

Let be independent and identically distributed random variables with distribution where θ is a real valued parameter. Suppose that and

where is continuously differentiable function with derivative for all θ. Show that the estimator obtained by solving the equation

where is consistent. Also derive its asymptotic distribution.

7.

Let be a sample from a distribution function with density . Since

one suggest the estimator

where

Find the condition on to guarantee that is a consistent estimate of . 8.

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