課程名稱：機率論
課程性質：必修
課程範圍：Section1.1~Section3.7
開課教師：姜志銘
開課學院：理學院
開課系級：應數二
考試日期(年月日)：
考試時限(Mins)：12:10~14:00
附註：
試題本文：
(12%) 1. You rolled two fair six-sided dice once and observe the total number of dots facing upward.
(a) What is the sample space?
(b) What is the probability of each sample outcome?
(c) What is the probability of E, the event that the outcome is even?
(14%) 2. A company has 3 machines A,B, and C for making parts. It has been observed that 80% of the parts produced by A are acceptable. Machine B and C producing 90% and 55% of acceptable parts, respectively. Each hour, machine A produce 300 parts, B produces 500 parts, and C produces 200 parts. All parts are mixed together at random in one bin and packed for shipment.
(a) What is the probability that the company ships a part that is acceptable?
(b) What is the probability that an acceptable part comes from machine A.
(14%) 3. An instant lottery ticket consists of a collection of boxes covered with gray wax. For a subset of the boxes, the gray wax hides a special mark. If a player scratches off the correct number of the marked boxes (and no boxes without the mark), then the ticket is a winner. Design an instant lottery game (less than 7 boxes) in which a player 3 boxes and the probability that a ticket is a winner is most close to 0.2.
(12%) 4. Each time a fisherman casts his line, a fish is caught with probability p, independent of whether a fish is caught on any other cast of the line. The fisherman will fish all day until 2 fishes are caught and then he will quit and go home. Let Ci denote the event that on cast i the fishermant catches the second fish.
Find P[C1], P[C2], P[C3], and P[Cn] as functions of p.
(18%) 5. (a) Give the definitions of Bernoulli (p), geometric(p), binomial(n,p) and Pascal (k,p) random variables.
(b) State the differences and equivalences between random variables Bernolli(p) and geometric(p).
(c) State the differences and equivalences between random variables binomial(n,p) and Pascal (k,p).
(d) State the differences and equivalences between random variables Bernolli(p) and binomial (n,p).
(e) State the differences and equivalences between random variables geometric(p) and Pascal (k,p).
(18%) 6. A tablet computer transmits a file over a wi-fi link to an access point. Depending on the size of the file, it is transmitted as N packets, where N had PMF
P(n)=c(1/2)^n n=1, 2, 3 / 0 o.w.
(a) Find the constant c.
(b) What is the probability that N is even?
(c) Each packet is received correctly with probability p, and the file is received correctly if all N packets are received correctly.
Find the probability that the file is received correctly.
(d) Find the CDF of N.
(e) Find E[N], E[N^2].
(12%) 7. The number of buses that arrive at a bus stop in T minutes is a Poisson random variable Bt with expected value T/6
(a) What is the PMF of Bt, the number of buses that arrive in T minutes?
(b) What is the probability that in a six-minute interval, three buses will arrive?
(c) What is the probability of nobuses arriving in a 12-minute interval?
(d) How much time should you allow so that with probability 0.99 at least one bus arrives?