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University Problem Set #8 ECE 313
of Illinois Page 1 of 2 Summer 2003
Assigned: Thursday, July 31
Due: Wednesday, August 6
Reading: Ross Chapters 6.5, 7.1-7.
Problems:
1. The number of hours R that a student spends r eading about probability in preparation for
the ECE 313 Final Examination and the number of hours S that the student spends
s leeping can be modeled as random variables with joint probability density function
f R , S
(x,y) =
K,10 ≤ x + y ≤ 20, x ≥ 0, y ≥ 0,
0,otherwise.
(a) What is the value of K?
(b) What is the marginal pdf of R?
(c) Unfortunately, the more the student tries to read about probability, the more confused the
student gets. Also, the less the student sleeps, the more tired the student gets. As a
result, the student’s percentage score T on the Final Exam is related to S and R via the
equation
T = 50 + 2.5( S – R ).
Find the pdf of T.
2. The random point ( X , Y ) is uniformly distributed on the shaded region shown below.
(a) Find the marginal pdf f X
(u) of the random variable X.
(b) Write down the marginal pdf f Y
(v) of the random variable Y from your answer to part
(b).
(c) Find P{ X < Y < 2 X }.
(d) What is f X | Y
(u|α), the conditional pdf of X given that Y = α, if α satisfies 0 < α < 1/2?
What is f X | Y
(u|α), the conditional pdf of X given that Y = α, if α satisfies 1/2 < α < 1?
Now, apply the theorem of total probability to compute the unconditional pdf of X from
f X | Y
(u|α). Do you get the same answer as in part (a)?
u
1
1
v
3. Let the random variables X and Y be independent and uniformly distributed on (0,1).
Find E(| X – Y |) and Var( X – Y ).
4. Let E[ X ] = 1, E[ Y ] = 4, var( X ) = 4, var( Y ) = 9, and ρ X , Y
(a) If Z = 2( X + Y )( X – Y ), what is E[ Z ]?
(b) If T = 2 X + Y and U = 2 X – Y , what is cov( T , U )?
(c) If W = 3 X + Y + 2, find E[ W ] and var( W ).
University Problem Set #8 ECE 313
of Illinois Page 2 of 2 Summer 2003
(d) If X and Y are jointly Gaussian random variables, and W is as defined in (c), what is
P{ W > 0}?
5. This problem has three independent parts. Do not apply the numbers from one part to the
others.
(a) If var( X + Y ) = 36 and var( X – Y ) = 64, what is cov( X , Y )? If you are also told that
var( X ) = 3•var( Y ), what is ρ X , Y
(b) If var( X + Y ) = var( X – Y ), are X and Y uncorrelated?
(c) If var( X ) = var( Y ), are X and Y uncorrelated?
6. Consider the random point ( X , Y ) of Problem 2 above.
(a) Compute E[ X ] and var( X ).
(b) Explain why the random variable Y has the same mean and variance as X.
(c) Compute E[ XY ] and hence find cov( X , Y ).
should hold. Is the above equation satisfied by the numerical values you obtained?
(d) The conditional pdf of X given Y = α was obtained in Problem 2 above, and it is easy to
see that the conditional pdf of Y given X = α is similar. Now, the best (least mean-
square error) estimate of Y given X = α is the mean of the conditional pdf of Y given X =
α. Thus, if X has value α ≤ 0.5, then
^
Y , the best estimate of Y , is 0.75 while if X has
value α > 0.5, then
^
Y = 0.5. Now, the best linear (least mean-square error) estimate of
Y (given that X is known to have value α) is
Y = a + bα where a and b were given in
class. Compute a and b, and draw a graph showing the estimates
^
Y and
Y as functions
of α. (Remember that 0 ≤ α ≤ 1). For what value(s) of α are the two estimates the same?
(e) Since the estimates
^
Y and
Y depend on the value of X , they really are functions of X ,
that is, they are random variables that can be expressed as
^
Y =
0.75,0 ≤ X ≤ 0.5,
0.5,0.5 < X ≤ 1
and
Y = a + b X. What are the average and the mean-square errors of each estimate? That is,
what are the values of E[( Y –
^
Y )], E[( Y –
Y )], E[( Y –
^
Y )
2 ], and E[( Y –
Y )
2 ]?
