ECEN/MAE 5513
Homework #7
1. The random variable X has a density function f X ( ) defined as

 ----fX ( ) =  2

0
<
Take one sample of the random variable - X1. a) Find and sketch the likelihood function. b) Find the maximum likelihood estimate for .
2. Using the sample of the random variable from problem 1, assume that
variable with the following density function.
1
 ----f ( µ ) =  µ2

0
a) Sketch f
X1 ( µ
µ
is a random
1
µ<1
) , b) Find the Bayesian maximum aposteriore estimate ˆ MAP .
3. Consider the measurement (Z) of a signal (X) in noise (V):
Z = X+V
1
fV ( µ ) = 
0
0 µ
else
1
1
fX ( ) = 
0
1
else
2
a) Sketch f V ( µ ) and f X ( ) , b) Sketch f Z X (
) vs.
fZ X(
1.5 ) vs.
) vs.
for
= 1.5 . d) Sketch f X Z (
for a fixed
. c) Sketch
. If the measurement
Z = 1.5 , find the Bayesian minimum mean square error estimate of the signal:
X̂ MS = E [ X Z = 1.5 ] .
4. The random variable X has the density function shown below. One sample, X1, of the
random variable is taken. Find the maximum likelihood estimate of the unknown constant . Show each step clearly.
fX ( )
--2-
5. Consider again the random variable X from problem 4. Assume now that is a random
variable, with prior density given below. Find the MAP estimate of , based on the single sample X1. Show each step clearly.
f (µ)
3
 3µ 2
f (µ) = 
0
µ
1
0<µ<1
else
6. Consider the measurement (Z) of a signal (X) in noise (V):
Z = X + V,
where V is Gaussian noise with zero mean and unity variance. The signal is
transmitted N times, and N independent measurements Z 1, Z 2, …, Z N are
received.
i.
Find the joint density function for the measurements,
f Z1, Z 2, …, Z N ( 1, 2, …, N ) .
ii.
Find the likelihood function L ( X ) .
iii.
Find the log likelihood function l ( X ) = ln L ( X ) .
iv.
Find the maximum likelihood estimate X̂ ML .
7. Assume now that the transmitted signal X from problem 6 is a Gaussian
2
random variable with mean X 0 and variance
. Assume that the same
measurements Z 1, Z 2, …, Z N are received.
i.
Find the Bayesian maximum aposteriore estimate X̂ MAP .
ii.
What does X̂ MAP approach as
2
0?
iii.
What does X̂ MAP approach as
2
?
iv.
What does X̂ MAP approach as N
?