課程名稱︰機率
課程性質︰系必修
課程教師︰陳文進
開課學院：電資學院
開課系所︰資工系
考試日期（年月日）︰4/25, 2007
考試時限（分鐘）：9:00 ~ 12:00
是否需發放獎勵金：是, 謝謝 :)
（如未明確表示，則不予發放）
試題 :
(1) (10%) Box A contains 8 items of which 3 are defective, and box B contains 5
items of which 2 are defective. An item is drawn at random from each box.
(a) What is the probability that both items are non-defective?
(b) What is the probability that onc item is defective and the other is
not?
(c) If one item is defective and the other is not, what is the probability
that the defective item came from box A?
(2) (10%) Three machines A, B and C produce respectively 60%, 30% and 10% of
the total number of item of a factory. The procetages of defective output
of these machines are repectively 2%, 3% and 4%. An item is selected at
random and is found defective. Find the probability that the item was
produced by machine C.
(3) (10%)
(a) If A and B are 2 events where P(A) = 0.9, P(B) = 0.8, show that
P(A∩B) ≧ 0.7.
(b) In general, for any 2 events A and B, show that
P(A∩B) ≧ P(A) + P(B) - 1
(4) (10%) Prove that if event A is independent of event B, and A is independent
of event C, and B∩C = ψ, then A is independent of B∪C.
(5) (10%) Let X be random variable having expected value μ and variance σ^2.
Find the expected value and variance of:
Y = (x - μ)/σ
(6) (10%)
(a) Determine the constant C so that P_k = P(X = k) is a probability mass
function of random variable X, where P_k = C(1/4)^k, k = 1, 2, 3, ...
(b) Find E(X) and Var(X).
(7) (10%) It is known that CD produced by a certain company will be defective
with probability 0.01, in dependently of each other. The company sells the
CD in packages of size 10 and offers a money-back guarantee that at most 1
if the CD in the package will be defective. If someone buys 3 packages,
what is the probability that he or she will return exactly 1 of them?
(8) (15%) People enter a gambling casino at a rate of 1 person for every 2
minutes.
(a) What is the probability that no one enters between 12:00 and 12:05?
(b) What is the probability at least 4 people enter the casino during that
time?
(9) (15%) Show that a geometric random variablehas memoryless property. That
is, if X is a geometric random variable, show analytically that
P{X = n + k | X > n} = P{X = k}
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