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%) ----------------------------------- -- 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: https://www.youtube.com/watch?v=QZjtO0oRBXk&feature=related 態度紀錄片: https://www.youtube.com/watch?v=TUmBz7z8qBo&feature=related --

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