課程名稱︰機率與統計
課程性質︰系訂必修
課程教師︰鍾嘉德 張時中 葉丙成 周俊廷
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰98.6.18
考試時限（分鐘）：原本是 2.5 小時，延長至 3 小時
是否需發放獎勵金：是
試題 :
1. (16%)
X1 and X2 are independent random variables, each following a
uniform distribution U(-1, 1). Let X_min = min(X1, X2) and
X_max = max(X1, X2).
(a) Are X_min and X_max independent? (Please show your answer based
on the definition of independence, not your intuition) (2%)
(b) Please find the joint CDF of X_min and X_max : F_Xmin,Xmax(x, y)
(8%)
(c) Please find E(X_max - X_min) (6%)
2. (12%)
K students participate in a party game where each student picks a
number from 1 to N uniformly and independently. The rule is that
whoever picks the same number as others is the winner. For example,
if David and Mary both pick 7, they are both winners. Let X be the
number of winners and Y is the number of those numbers being picked
by the winners.
(a) Are X and Y independent? (Please show your answer based on the
definition of independence, not your intuition) (2%)
(b) Please find E(X) and E(Y) (10%)
原 PO 註：假設 8 個人選的數字是 3, 4, 4, 5, 5, 5, 6, 6
則 X = 7, Y = 3
3. (13%)
X and Y have a joint PDF as follows:
f_X,Y(x, y) = exp(-2 [x^2-xy+y^2] / 3) / (sqrt(3)pi)
(a) Are X and Y independent? (2%)
(b) Let V = aX + bY and W = cX + dY. Please find a, b, c, and d such
that V and W are independent random variables. (6%)
(c) Z = X - Y. Derive f_z. (5%)
(Hint: For a real-valued symmetric matrix A, we can find that A=UDU'
where D is a diagonal matrix with its element being the eigenvalues
of A, and U is a uniary matrix with columns that are n orthonormal
eigenvectors of A.)
4. (21%)
David lives near the area where 3 MTA lines merge as shown in the
figure below. Based on his experience, he finds out
i. The trains arrive in the blue-line, red-line and green-line stations
following Poisson distribution, each with a rate of 1/4, 1/2,
and 1/4 per minute, respectively.
ii. When a train arrives at any one of the above three stations, the
probability that the train has an empty seat is 0.5.
iii. Whenever a train arrives in any of the three stations, there will
be K people (excluding David) waiting for the train. K is
uniformly distributed between 0 and 4. However, more people are
waiting in the main station. Therefore, K (again, excluding
David) is uniformly distributed between 0 and 16 for the main
station.
iv. Every one waiting for the train has an equal probability to get the
empty seat.
blue-line -----o----------
|
read-line -----o---------Main station-------NTU--------
|
green-line -----o----------
(o is station).
David can take the train at any train station to go to NTU. However,
he does not like to wait nor stand in a train. Let's find a station
for him so that E(waiting time) * Prob(standing in a train) is
minimized.
(a) When David waits in the main station, is the probability that
three is an empty seat still 0.5? If not, what is the
probability that there is a seat left? (2%)
(b) When waiting in the main station, what is the probability that
David will get an empty seat? (2%) (Note that we assume that
nonegets off the train in the main station)
(c) Show that the total number of train arrivals in one period of
time is still Poisson and the total rate is 1 per minute. (7%)
(Hint: Use moment generating function)
(d) When waiting in the main station, what is the distribution of
inter-arrival time (i.e., the time between two consecutive train
arrivals)? (6%) (Hint: the time interval between two
consecutive events that follow a Poisson distribution follows
an exponential distribution)
(e) Given your above answers, which station should David choose to
achieve his objective. (4%)
5. (14%)
A semiconductor wafer has M VLSI chips on it and these chips have the
same circuitry. Each VLSI ship consists of N interconnected transistors.
A transistor may fail (not function properly) with a probability p
because of its fabrication process, which we assume to be independent
among individual transistors. A chip is considered a failure if there
are n or more transistor failures. Let K be the number of failed
transistors on a VLSI chip, which is therefore a random variable (R.V.)
(a) What is a random variable? (4%)
(b) What is the sample space (also called outcome set) over which
R.V. k is defined? (2%)
(c) Let X_i = 1 if a chip i fails and X_i = 0 if a chip i is good.
Derive the probability that a chip is good, i.e.,
p_g = Pr{X_i = 0} = ? (3%)
(d) Whether one chip is good or fails is independent of other chips.
Let the yield of a wafer be defined as the percentage of good
chips in the wafer, i.e.,
M
Y = (1 - 1/M Σ X_i) ×100%
i=1
Then derive μ_Y = E[Y] = ? (σ_Y)^2 = Var[Y] = ?
(Hint: Utilize p_g obtaind form 5.(c)) (5%)
6. (24%)
You are observing a radar signal sequence
Y_k = θ+ω_k, k = 1, 2, 3...
where θ is an unknown constant, and ω_k is N(μ,σ^2) and
independent and identical over time index k.
(a) Now you have one observation Y_1 = x, where x is an observed value.
And you set an estimate of θ as Θ = x - ω_1. Derive
f_(Θ|Y_1) (s | x), E[Θ|x] and Var[Θ|x]. (8%)
(b) Given N observations of Y_k, how do you estimate the value of θ?
(3%) Is your estimate biased or unbiased, why? (5%)
(c) When N = 100, propose an approximation method of your confidence
(in terms of probability) that your estimate in (6.a) is within
0.2σ from the true parameter θ. Please explain quantitatively
why. (8%)
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真的很難..
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