UCSD ECE153
Handout #36
Prof. YoungHan Kim
Thursday, May 26, 2011
Final Examination (Spring 2008)
1.
Coin with random bias (20 points).
You are given a coin but are not told what its bias
(probability of heads) is. You are told instead that the bias is the outcome of a random
variable
P
∼
Unif[0
,
1]. Assume
P
does not change during the sequence of tosses.
(a) What is the probability that the first three flips are heads?
(b) What is the probability that the second flip is heads given that the first flip is
heads?
2.
Estimation (20 points).
Let
X
1
and
X
2
be independent identically distributed random
variables. Let
Y
=
X
1
+
X
2
.
(a) Find
E
[
X
1

X
2

Y
].
(b) Find the minimum mean squared error estimate of
X
1
given an observed value of
Y
=
X
1
+
X
2
. (Hint: Consider
E
[
X
1
+
X
2

X
1
+
X
2
].)
3.
Stationary process (20 points).
Consider the Gaussian autoregressive random process
X
k
+1
=
1
3
X
k
+
Z
k
,
k
= 0
,
1
,
2
, . . . ,
where
Z
0
, Z
1
, Z
2
, . . .
are i.i.d.
∼
N
(0
,
1).
(a) Find the distribution on
X
0
that makes this a stationary stochastic process.
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 Spring '11
 Kim
 Signal Processing, Probability theory, Stochastic process, Autocorrelation, Stationary process

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