課程名稱︰機率導論
課程性質︰數學系大二必修
課程教師︰鄭明燕
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014/4/17
考試時限（分鐘）：13：20～15：20
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. (10%) A closet contains 12 pairs of shoes. If 10 shoes are randomly
selected, what is the probability that there will be exactly 2 complete
pairs?
2. Consider two boxes, one containing 3 black and 2 white marble, and the
other containing 2 black and 1 white marble. A box ix selected at random,
and a marble is drawn at random from the selected box.
(a) (10%) Find the probability that the marble is black.
(b) (10%) Find the probability that the first box was the one selected,
given that the marble is white.
3. Experience shows that 10% of the people reserving tables at a certain
restaurant never show up. Suppose the restaurant has 30 tables and takes 32
reservations. Let X be the number of reserved tables that are actually
taken.
(a) (10%) What is the probability distribution of X?
(b) (10%) Find the probability that the restaurant will be able to
accommodate everyone.
4. Suppose 2% of the blue fuses and 1% of the red fuses produced by a company
are defective. And 100 fuses of the same color are packed in one box for
sale. If you buy one box of blue fuses and one box of red fuses. Let X
donote the number of defective you get.
(a) (10%) Use the Poisson approximation to P(X≦2).
(b) (10%) Use the Normal approximation to P(X≦2).
5. (10%) Suppose X is a random variable distributed as Poisson(λ), where
-λ k
λ > 0, i.e. P(X = k) = e λ / k!, k = 0,1,2,... Find the expectation of
1
—————.
(X+1)(X+2)
6. Let X be a random variable with cumulative distribution function
x
︴ 0.5e , if x < 0,
F(x) = ︴
︴ -x
︴1 － 0.5e , if x ≧ 0.
(a) (10%) What is the probability density function of X?
2
(b) (10%) What is the probability density function of X ?