臺灣大學數學系
八十六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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(10 points) Show that if ; are independent and identically
distributed random variables with . Find the
limiting distribution of the sequence of random variables 1.
(10 points) Let be a sequence of random variables satisfying
Then show that converges to in probability implies that converges to almost surely.
2.
(10 points) Consider an urn which contains black and white balls which are well mixed. Repeated drawings are made from the urn, and after each drawing the ball drawn is replaced, along with balls of the same color. Here is an integer.
Let , and be the proportion of black balls in the urn after the th draw. Find the probability that the th ball drawn is black.
3.
(20 points) Let be a sample from uniform where , are unknown.
Show that is sufficient. (10
points) 1.
Assuming that is complete find a U.M.V.U. estimate of . (10 points)
2.
4.
(15 points) Suppose is a sample from a population with density
where is the standard normal density and
. 5.
Show that maximum likelihood estimates do not exist, but that
if, and only if, equals one of the numbers . Here . (10 points)
1.
Give a brief explanation why (a) holds. (5 points) Hint: continuous pdf versus discrete pdf
2.
(15 points) The normally distributed random variables are said to be serially correlated if we can write
where and are independent random variables. Derive the likelihood ratio test of (independence) versus (serial correlation).
6.
(20 points) Assume that for , where are
independent random variables ( ), and are iid ,
and the s and s are independent. Statistician A proposes to estimate β by
, denoted it by , since is normally distributed with mean .
Statistician B proposes to estimate β by , denoted it by , since follows a linear regression model if s are viewed as constants. For these two estimates and , which one will you use? Please give reasons to support your conclusion.
7.
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