ps5_solution_W19 University of California, San Diego ECE 250
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Course
ECE 250
Institution
ECE 250
UNIVERSITY OF CALIFORNIA, SAN DIEGO
Electrical & Computer Engineering Department
ECE 250 - Winter Quarter 2019
Random Processes
Solutions to P.S. #5
1. Cauchy–Schwartz inequality.
(a) Prove the following inequality: (E(XY ))2 ≤ E(X2
)E(Y
2
). (Hint: Use the fact that for
any real t, E...
san diego electrical amp computer engineering department ece 250 winter quarter 2019 random processes soluti
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UNIVERSITY OF CALIFORNIA, SAN DIEGO
Electrical & Computer Engineering Department
ECE 250 - Winter Quarter 2019
Random Processes
Solutions to P.S. #5
1. Cauchy–Schwartz inequality.
(a) Prove the following inequality: (E(XY ))2 ≤ E(X 2 )E(Y 2 ). (Hint: Use the fact that for
any real t, E((X + tY )2 ) ≥ 0.)
(b) Prove that equality holds if and only if X = cY for some constant c. Find c in terms of
the second moments of X and Y .
(c) Use the Cauchy–Schwartz inequality to show the correlation coefficient |ρX,Y | ≤ 1.
p p p
(d) Prove the triangle inequality: E((X + Y )2 ) ≤ E(X 2 ) + E(Y 2 ).
Solution:
(a) We have, for every t ∈ R, E[(X + tY )2 ] ≥ 0, i.e., min t2 E[Y 2 ] + 2tE[XY ] + E[X 2 ] ≥ 0.
t∈R
From calculus, we see that the expression on the left attains its minimum value when
E[XY ]
t=− , and this minimum value is given by
E[Y 2 ]
(E[XY ])2
E[X 2 ] − . Thus, since E[Y 2 ] is non-negative, we have
E[Y 2 ]
E[X 2 ]E[Y 2 ] − (E[XY ])2 ≥ 0, i.e.
(E[XY ])2 ≤ E[X 2 ]E[Y 2 ].
(b) For the “if” part, we see that if X = cY for some constant c, then E[X 2 ] = c2 E[Y 2 ] and
E[XY ] = cE[Y 2 ], thus
2. Neural net. Let Y = X + Z, where the signal X ∼ U[−1, 1] and noise Z ∼ N(0, 1) are
independent.
(a) Find the function g(y) that minimizes
MSE = E (sgn(X) − g(Y ))2 ,
where (
−1 x ≤ 0
sgn(x) =
+1 x > 0.
(b) Plot g(y) vs. y.
Solution: The minimum MSE is achieved when g(Y ) = E(sgn(X) | Y ). We have
Z ∞
g(y) = E(sgn(X) | Y = y) = sgn(x)fX|Y (x|y) dx .
−∞
To find the conditional pdf of X given Y , we use
(1
fY |X (y|x)fX (x) 2 −1 ≤ x ≤ 1
fX|Y (x|y) = , where fX (x) =
fY (y) 0 otherwise.
2
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