[試題] 104上 張勝凱 統計學上 第一次期中考
課程名稱︰統計學與計量經濟學暨實習上
課程性質︰必修
課程教師︰張勝凱
開課學院：社科院
開課系所︰經濟系
考試日期（年月日）︰104.10.16
考試時限（分鐘）：180 min.
試題 :
Problem 1. (30 points(6,6,8,10))
An electronic fuse is produced by five production lines in a manufacturing
operation. The fuses are shipped to supplier in 100-unit lots. All five
production line production lines produce fuses at the same rate and normally
produce only 2% defective fuses, which are dispersed randomly in the outputs.
Unfortunately, production line 1 suffered mechanical difficulty and produced
5% defective fuses during the month of October. A customer received a lot
produced in October and tested three fuses, one failed, denoted as the event
A. Let B denote the event that a fuse was produced from production line 1.
(a) Given that the lot was produced in production line 1, what is the
probability for the event A, that is P(A|B).
(b) Given that the lot was produced in one of the four other production lines,
c
what is the probability for the event A, that is P(A|B ).
(c) What is the unconditional probability of event A, that is P(A).
(d) What is the probability that the lot was produced on production line 1,
given the event A.
Problem 2. (20 points (5,5,10))
Part one: Consider the following events in the toss of a fair die:
A: Observe an odd number.
B: Observe an even number.
C: Observe 5 or 6.
(a) Are A and B independent events?
(b) Are A and C independent events?
Part two: Three brands of tea, X, Y, and Z are to be ranked according to taste
by a judge. Define the following events:
A: Brand X is preferred to Y.
B: Brand X is ranked best.
C: Brand X is ranked second best.
D: Brand X is ranked third best.
(c) If the judge actually has no taste preference and randomly assigns ranks
to the brands, is event A independent of event B, C, and D?
(Hint: if A is independent of B, then P(A|B) = P(A).)
Problem 3. (30 points (8,10,12))
The joint probability density function (pdf) of X and Y is as follow.
╭
│ x + y , if 0 ≦ x ≦ 1 ; 0 ≦ y ≦ 1
│
f (x,y) = ﹤
XY │
│ 0 , otherwise.
╰
(a) Is f a legitimate joint pdf function? Why or why not.
XY
A single draw is made for Y and you are asked to predict the value of Y
2
(b) Find the best constant predictor c, in the sense minimizing E(U ) where
U = Y - c.
Now, a single draw of random vector (X,Y) is made and you are told the value
of X that was drawn and asked to predict the value of Y that accompanied it.
2
(c) Find the best predicor h(X), in the sense minimizing E(U ) where U = Y-h(X)
and h(X) is any function of X.
Problem 4. (20 points (6,6,8))
Let X be a random variable with a probability density function
╭
│ c(2-x) , 0 ≦ x ≦ 2
│
f (x) = ﹤
X │
│ 0 , otherwise
╰
(a) Find c.
(b) Find the cumulative distribution function of X, F (r) is a known constant.
X
(c) Find P(1≦X) and P(X=1).
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