程序代写案例-ECE 153

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ECE 153
Spring 2018
Problem 1: Let X and Y be two independent random variables according to the following densities.
fX(x) =
{
e−x if x
≥ 0,
0 otherwise,
fY (y) =
{
1 if 0 ≤ y ≤ 1,
0 otherwise.
(1)
(a) Find the pdf of U = max{X,Y }.
(b) Find the conditional pdf of X given the event {X ≥ Y }.
(c) Find the joint cdf of X and U , FU,X(u, x).
1
2
Problem 2: Let X and Y be two random variables with joint pdf
f(x, y) =
{ xy
c , 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ x+ y ≤ 1
0 otherwise
(2)
(a) Find the value of c.
(b) Find the conditional pdf fX|Y (x|y).
(c) Find the MMSE estimator of X given Y .
(d) Find the CDF of Z = E(X|Y ) in terms of CDF of Y .
(e) Find the MMSE linear estimator of X given Y .
3
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Problem 3: Let Xn = αZn−1 + βZn+1 for n ≥ 1, where Z0, Z1, Z2, . . . are i.i.d. ∼ N (0, 1). Let
Yn = Xn +Wn, where Wn are i.i.d. N (0, 1), independent of Xn.
(a) Is the process {Xn} Gaussian? Justify your answer.
(b) Find the mean and autocorrelation functions of Xn.
(c) Find autocorrelation functions of Yn and cross correlation function RXY .
(d) Is Xn wide-sense stationary? Justify your answer.
(e) Is Xn strict-sense stationary? Justify your answer.
(f) Find MMSE estimator of Xn given Xn−2.
(g) Find MMSE estimator of Xn given Yn.
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Some Useful Facts:
• ∫ 10 y(1− y)4dy = 130
• ∫ 10 y3(1− y)2dy = ∫ 10 y2(1− y)3dy = 160
• ∫ 10 y2(1− y)2dy = 130
• ∫ x0 te−tdt = 1− (x+ 1)e−x
• The best linear MMSE estimate is
Xˆ =
Cov(X,Y )
σ2Y
(Y − E(Y )) + E(X).
• Let X(t), t ∈ R, be a WSS process input to a LTIS with impulse response h(t):
X(t) Y (t)h(t)
If the system is stable, then the input X(t) and output Y (t) are jointly WSS with:
1. E(Y (t)) = H(0)E(X(t))
2. RY X(τ) = h(τ) ∗RX(τ)
3. RY (τ) = h(τ) ∗RX(τ) ∗ h(−τ)
RX(τ) RY (τ)h(τ) h(−τ)
RY X(τ)
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