課程名稱︰通訊隨機過程
課程性質︰選修
課程教師︰鐘嘉德
開課學院：電資
開課系所︰電信所 電機所
考試日期（年月日）︰
考試時限（分鐘）：3 HR
是否需發放獎勵金：是
試題 :
1. Determine whether each of following statement is true or false.
If the statement is true, prove it. If the statement is false, give a
counterexample or explanation. Corrent choice without any proof,
counterexample or explanation is not acceptable.
(a) If two events A and B are mutually exclusive(i.e. their intersection is
an empty set), then A and B are independent.
(b) If the probability of an event A is equal to one, then A must be the
universe space.
(c) If three random variables X(μ), Y(μ) and Z(μ) are independent in pairs
(i.e. any two of them are statistical independent), then they are mutually
independent.
(d) The function R(τ)= |τ|exp(-|τ|) can be the autocorrelation function
od a wide-sense stationary random process
(e) The function S(ω)= exp[ω +2ω^2 -ω^4] can be the power spetrum of
a wide-sense stationary random process.
(f) If a wide-sense stationary random process X(μ,t) has nonzero delta
spetrum on ω=0, then the mean of X(μ,t) must be nonzero.
(g) If X(μ,t) is a real-valued wide-sense stationary Gaussian random process
with mean η_x = 1 and the autocorrelation function Rx(τ)= sin(τ)/τ + 1
,then the random variable X(μ,t0) and X(μ,t0 +π) are independent for
any fixed t0.
(h) If the input to a linear and time-invariant system is a Gaussian random
process, then the output is a stationary Gaussian random process.
2. Consider the experiment of rolling two fair dice independently. Define
two random variable X(μ) and Y(μ) as the value of both dice that face
upward after a single trail. Also, define the random variable Z(μ)=
X(μ)+Y(μ) and W(μ)=X(μ)Y(μ).
(a) Determine the probability Pr(Z(μ)=n) for all integer n.
(b) Determine the conditional probability Pr(Z(μ)=n|X(μ)=m) for all
integers n and m
(c) Determine the mean E(W(μ))
(d) Determine the variance Var(W(μ))
3. Define the real-valued random process X(μ,t) by
X(μ,t) = R(μ)cos(ωt +ψ(μ)) where R(μ) and ψ(μ) are independent
random variable, E(R^2(μ)) < ∞, and ψ(μ) is uniformly distributed
in the interval (0,2π). Derive the mean function and the autocorrelation
function of X(μ,t), and determine whether X(μ,t) is wide-sense stationary
or not.
4. Consider a linear and time-invariant system with impluse response h(t),
input process X(μ,t) and output process Y(μ,t). Show that if h(t)=0
outside the time interval (0,T) and X(μ,t) is zero-mean white noise,
then Ry(t1,t2)=0 for |t1-t2|>T.
5. Consider independent and identically distributed random variable X1(μ),
X2(μ),....Xn(μ) which are marginally uniform in the interval (0,1).
Show that if Y(μ)=max(X1(μ),X2(μ),...,Xn(μ)), then Fy(y)=y^n for
0<y<1, Fy(y)=0 for y<0 and Fy(y)=1 for y>1.
6. Consider the wide-sense stationary random process X(μ,t) and Y(μ,t) which
are related by Y(μ,t)= X(μ,t+1) -X(μ,t-1). Express
(a) Ry(τ) in terms of Rx(τ) and
(b) Sy(ω) in terms of Sx(ω).
7. Suppose that X1(μ) and X2(μ) are independent Poisson random variables
with parameters λ1 and λ2, i.e
Pr(Xi(μ)=k) = exp(-λi)(λi)^k / k! , if k is a nonnegative integer
0, otherwise
for i=1,2. Define a new random variable Y(μ)= X1(μ) +X2(μ).
Prove that Y(μ) is a Poisson random variable with parameter λ1+λ2.
8. Consider the real-valued stationary Gaussian random process X(μ,t)
which has mean zero and autocorrelation Rx(τ)= exp(-|τ|)˙cos(τ).
Also let Y(μ,t) = X^2(μ,t). Derive the autocorrealtion function
of Y(μ,t).
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