课程名称︰机率
课程性质︰必带
课程教师︰刘长远
开课学院：电资学院
开课系所︰资讯工程系
考试日期（年月日）︰2015/6/22
考试时限（分钟）：180分钟
试题 :
1. Poisson Process
1.(15 pts) Customers arrive to a restaurant according to a Poisson process
with rate 10 customers/hour. The restaurant opens daily at 11:00 am.
Find the following:
(a) What is the probability that there are 15 customers in the restaurant
at 1:00pm, given that there were 12 customers in the restaurant at
12:50pm.
(b) Given that a new customer arrived at 11:13am, what is the expected
arrival time of the next customer?
(c) If a customer arrive to restaurant at 2:00pm, what is the probability
that the next four customers will arrive before 3:00pm?
2.(10 pts) Transmitters A send messages to a receiver in a Poisson process
with average message arrival rates of λ_A. All messages are so brief that
we may safely assume that they occupy only single points in time. The
number of words in every message, regardless of its transmitting source,
may be considered to be an independent experimental value of random
variable W with PMF p_w(1) = 1/3, p_w(2) = 1/2, p_w(3) = 1/6 and p_w(w) = 0
otherwise.
(a) Determine the PDF for T, the time from t = 0 until the receiver has
received exactly eight three-word messages from transmitter A.
(b) Independent of what happens to all other words, a transmitter damades
any particular word it sends with probability 10^-3. What is the
probability that any particular damaged word is part of a three-word
message?
2. Markov Chain Property
1.(25 pts) Days are either good(G), fair(F),or sad(S). Let F_n, for instance
, be the event that the nth day is fair. Assume that the probability of
having a good, fair, or sad day depends only on the condition of the
previous day as dictated by the conditional probabilities
P( F_n+1 | G_n ) = 1/3, P( F_n+1 | S_n ) = 3/8, P( S_n+1 | F_n ) = 1/6
P( F_n+1 | F_n ) = 2/3, P( S_n+1 | G_n ) = 1/6, P( S_n+1 | S_n ) = 1/2
Assume that the process is in the steady state. A good day is worth $1,
a fair day is worth $0, and a sad day is worth $-1
(a) Use G, F, S to denote respectively the states good, fair, and sad.
Please draw the Markov Chain for this problem.
(b) Determine the expected value of a randomly selected day.
(c) With the process in the steady-state at day zero, we are told that
the sum value of days 13 and 14 was $0. What is the probability
that day 13 was a fair day?
3. Gaussian Distribution
1.(15 pts) Your company is producing special battery pascks for the most
popular toy during the holiday season. The life span of the battery pack
is known to be normally distributed with a mean of 250 hours and a
standard deviation of 20 hours.
(a) What percentage of battery packs lasts longer than 260 hours?
(b) If a simple random sample of four battery packs is selected from your
company, what is the probability that the average of these four packs
is longer than 260 hours?
2.(10 pts) An insurance company sells %10,000 car insurance policies with
a one year term for a premiun of $400. If the insured involves in a car
accident within one year of the start of the policy, the insurance company
has to pay $10,000 to the policy's beneficiary. The probability of a
person detting into a car accident within one year is 1/10. If the company
sells 100 such policies in one year, what is the probability that the
company loses money at the end of the year after paying out any amounts
it has to pay out?
4. Statistics
1.(10 pts) We are sure that the individual grades in a class are normally
districuted about a standard deviation σ of 20 and have mean μ equal
to either 70 or 80. Consider a hypothesis test of the null hypothesis
H_0(μ = 80) with a statisic which is the experimental value of a single
grade. We wish to test H_0 at the 0.01 level of significance.
(a) Which one is more appropriate to use, rifht-sided, left-sided, or
two-sided test? Briefly explain your answer.
(b) Determine the conditional probability of false acceptance of H_0.
(c) Determine the conditional probability of false rejection of H_0.
2.(8 pts) The Geometric distribution with probability P has pdf
f(x) = ((1-P)^(x-1))‧P, x = 1,2,3,...
Suppose the data 5, 2 , and 3 were drawn independently from such a
distribution.
Find the maximum likelihood estimate (MLE) of P.
3.(7 pts) The pdf of the population distribution is
f(x) = (1 + ax)/2 , -1 <= x <= 1
-1 <= a<= 1
f(x) = 0 , otherwise
and the estimator for a is given by a* = 3X'
(where X' is the simple mean of X_1, ..., X_n)
Is a* a biased estimator for a?
[Hint:E(X') = (1/n)‧(E(X_1) + E(X_2)+...+E(X_n)) = E(X)]
※ 注:考卷上有附normal distribution table