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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