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/電機系 考試日期（年月日）︰100/10/11 考試時限（分鐘）：50分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : PROBLEM 1 (15%): Consider a random variable X with an unknow expectation E[x].To estimate E[X], we take N independent measurements of X , Xi,i=1,2,3,4,5,...,N.Assume that each Xi is modeled by Xi = E[X]+Wi , where Wi has mean = 0 and variance = 1. Let the sample mean SN of Xi,i=1,2,3,4,5,...,N be adopted as an estimate of E[X].Find the number of measurements Xi required for making that the probability of |SN-E[X]|<=1 is at least 0.99 . PROBLEM 2 (20%) : Conside that a random variable Θ is uniformly distributed in (-π,+π).We create two random variables X and Y as follows : X=cos(Θ) and Y=sin(Θ). (a)Find the probability density function(PDF) for each of X and Y. (b)Are X and Y independent ? Why? (7) (c)Are X and Y uncorrelated ? Why? (5) PROBLEM 3 (15%) : Consider that an observed random variable Y and a desired random variable X are related by Y = X + W , where X and W are two independent Gaussian random variables with mean zero and different variances (σx)^2 and (σw)^2, respectively. (a)Find the correlation coefficient of X and Y as a function of the variances of X and W.(10) (b)Find the PDF of Y. PROBLEM 4 (25%): Consider that the output of a linear-invariant(LTI) system is given by y(t)={exp{-t}-exp{-2t}}u(t) when the input is x(t)=exp{-3t}u(t),where u(t) is the unit step function. (a)Find the transform function H(s) of the LTI system.(7) (b)Find the region of convergence of H(s).(6) (c)Is the LTI system causal ? Why?(6) (d)Is the LTI system stable ? Why?(6) PROBLEM 5 (25%): Consider that the output of a linear-invariant(LTI) system is given by y[n]={9(1/2)^n + 10(1/3)^n}u[n] when the input is x[n]=[(1/6)^n]u[n],where u[n] is the unit step function. (a)Find the transform function H(z) of the LTI system.(7) (b)Find the region of convergence of H(z).(6) (c)Is the LTI system causal ? Why? (6) (d)Is the LTI system stable ? Why? (6) ----------------------------------------------------------------------------- --

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