課程名稱︰隨機信號與系統
課程性質︰選修
課程教師︰李枝宏 教授
開課學院：電資學院
開課系所︰電信所/電機所
考試日期（年月日）︰100/11/8
考試時限（分鐘）：2 HR
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Problem 1
Consider three real jointly Gaussian random variables X1, X2, X3 with common
variance σ^2. Moreover, the expection are m1,m2 and m3, respectively.
The covariance Matrix of the three variables id given by:
┌ ┐
│σ^2 ρ12 ρ13 │
Cx = │ρ21 σ^2 ρ23 │
│ρ31 ρ32 σ^2 │
└ ┘
Now we perform a linear transformation using Yi=A(Xi-B), i=1,2,3 to create
three random cariables Y1,Y2 and Y3. A and B are two constants.
(a) Are Y1, Y2 and Y3 jointly Gaussian? WHY? (7%)
(b) Find the jointly probability density function of X1 and X2, You
must justify yoru answer.(5%)
(c) Find the jointly PDF of Y1,Y2 and Y3. You must justify your answer.(8%)
---------------------------------
Problem 2
Consider the problem of measuring some statistial quantity. Ley Xi be the ith
measurements of the length (in cm) of an IC chip whoes actual length is B
in cm. Assume that each independent measurements Xi is modeled by the form:
Xi= B+Ei, where the measurement Ei is a random veriable with zero mean and
variance 1 in cm^2.
After taking N independent measurements, we use the sample mean Sn of Xi,
i=1,2,....n. to estimate the real length B in cm.
(a) Is Sn an unbiased estimator of B? WHY? (5%)
(b) Find the variance of Sn. You must justify your answer. (5%)
(c) Is Sn a consistent estimator of B? WHY? (5%)
(d) Find the number n of independent measurements required for the
Chebyshev's inequality to guarantee that Sn is within 0.1 cm of
the exact length B with a probability >= 0.99. You must justify
your answer. (5%)
----------------------------
Problem 3
Consider a sequence of independent identical distribution Gaussian random
variable Zi, i=1,2,... with zero mean and variance 1. we create a random
process S[n] = (Zn + Zn-1)/2 with S[0]=0.
(a) Find the mean of S[n]. You must justify your answer.(5%)
(b) Find the autocorrelation of s[n]. You must justofy your answer.(7%)
(c) Find the correlation coefficient of S[k] and S[i]. (8%)
--------------------------------
Problem 4
Consider the problem that an observation of transitting a wide-sence
stationary (WSS) random signal X[n] throught a channel is given by
n
Y[n] = (-1) X[n]. Let the autocorrelation function of X[n] be Rxx[k],
where k denotes the time difference.
(a) Is Y[n] WSS ? WHY ? (10%)
(b) Are X[n] and Y[n] jointly WSS? WHY? (10%)
---------------------------------
Problem 5
Consider that a random process X(t) has independent increments with X(0)=0.
Assume that the increment X(t2)- X(t1) in the time interval [t1,t2] is a
Poission random variable with expection given by λ(t2-t1), where λ>0
denotes the average rate.
(a) Find the joint PDF of X(t2) and X(t1). You must justify your answer.(6%)
(b) Find the autocorrelation function of X(t). You must justify your answer
(7%)
(c) Is X(t) mean-ergodic ? WHY? (7%)
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--
因為我沒有學歷，我也沒有背景，我也沒有有錢的老爸老媽。
能離成功比較接近，就是「態度」
《陳建州 台啤態度紀錄片》
態度預告片: http://www.youtube.com/watch?v=QZjtO0oRBXk&feature=related
態度紀錄片: http://www.youtube.com/watch?v=TUmBz7z8qBo&feature=related
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