ECE6303
HW 6
Fall 2018
1. X and Y have joint density function
f ( x, y) 
XY
1 xy
A
, | x | 1,| y | 1; zero,otherwise.
(1) Find A so that the above defined is a valid joint density function.
(2) Are X,Y uncorrelated? Show your answer.
(3) Are X,Y independent? Show your answer.
(4) Find the joint density function of X 2 and Y 2
(5) Prove or disprove that X 2 and Y 2 are independent.
(Video is required)
2.
f ( x, y)  B, x2  y 2  1; zero, otherwise.
XY
(a) Find B so that the above defined function is a valid joint pdf.
(b) Find E{X }, E{X | y}, E{X |Y}, E{Y}, E{Y | x}, E{Y | X } and E{XY} .
(c) Prove or disprove X and Y are uncorrelated.
(Video is required)
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3. X and Y are i.i.d. random variables with common p.d.f.
f (x) exu(x), f (y) eyu(y).
X
Y
f X ( x)
f Y ( y)
x
y
(a) Z  X Y . Find fZ ( z) by characteristic function method.
(b) Z  XY . Find fZ ( z) by auxiliary variable method.
4. X and Y are independent with exponential densities
f X ( x)  e xu( x), fY ( y)  e yu( y).
Find the joint density Z=X+Y and W=X/Y.
5.
Suppose the density function of X is an even function. Prove or disprove
that X and |X| are uncorrelated. Prove or disprove that X and |X| are
independent.
6. Suppose X and Y are independent normal random variables. Let
2
 Z  X   Y
(where  ,  are nonzero constants)

W  X   Y
Given  2 x2   2 y2 . Prove or disprove that Z and W are independent
normal random variables.
7.
Prove that for any X, Y real or complex
(a) | E{ XY }|2  E{| X |2}E{| Y |2}
(b) (triangle inequality) E{| X Y |2}  E{| X |2}  E{|Y |2}.
8. X , Y and Z are i.i.d. random variables uniform in (0, 1). Find E{
X
}.
X Y  Z
9. Let X and Y be i.i.d. random variables with common probability mass
function P( X  k )  P(Y  k )  pqk , k  0,1,2,...
Define Z  min{X ,Y }. Find the probability mass function of Z.
3
 Axy, 0 x y 1,
 0, otherwise.
10. X and Y have joint density function f ( x, y)  
XY
(a) Find A.
(b) Find and sketch the marginal density functions f ( x) and f ( y) .
(c) Prove or disprove that X,Y are independent.
(d) Find E{X }, E{Y} and E{XY } .
(e) Prove or disprove that X,Y are uncorrelated.
(f) Find and sketch the conditional density functions f ( y | x) and f ( x | y) .
(g) Find E{Y | x}, E{Y | X } and E{X | y}, E{X |Y}.
X
Y
 Ay 2 , 0 x| y|1
11. Given the joint probability density function f XY ( x, y)  
 0, otherwise.
4
1
y
fXY ( x, y)
yx
1 x
x
y  x
1
y
1
1
(a) Find A. Find E{X }, E{Y} and E{XY } .
(b) Prove or disprove that X,Y are uncorrelated and/or independent.
(c) Find and sketch the conditional density functions f ( y | x) and f ( x | y) .
(d) Find E{Y | x}, E{Y | X } and E{X | y}, E{X |Y}. Sketch E{X | y}.
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