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University of Illinois Spring 2007
ECE 413: Final Examination
Tuesday May 8, 2007 1:30 p.m. — 4:30 p.m. 269 Everitt Laboratory
Name:
Section: 2 C, 10 am 2 D, 11 am
University ID Number:
Signature:
Instructions
This exam is closed book and closed notes except that three 8.5”×11” sheets of notes are permitted: both sides may be used. Calculators, laptop com- puters, PDAs, iPods, cellphones, e-mail pagers, headphones, etc. are not allowed. The exam consists of 7 problems worth a total of 225 points. Note that the problems are not weighted equally and pace yourself accordingly. Write your answers in the spaces provided. SHOW YOUR WORK. Answers without appropri- ate justification will receive very little credit. If you need extra space, use the back of the previous page.
Grading
25 points
25 points
24 points
36 points
24 points
56 points
35 points
Total (225 points)
- [25 points] Suppose the CDF of a continuous random variable is as shown below.
1 / 2
u
FX (u) 1
− 1 0 1 2
(a) [5 points] Find P {|X| ≤ 0. 8 }.
(b) [8 points] Find E[X].
(c) [12 points] Find var(X).
- [24 points] A particular webserver may be working or not working. If the webserver is not working, any attempt to access it fails. Even if the webserver is working, an attempt to access it can fail due to network congestion beyond the control of the webserver. Suppose that the a priori probability that the server is working is 0.8. Suppose that if the server is working, then each access attempt is successful with probability 0.9, independently of other access attempts.
(a) [6 points] Find P [ first access attempt fails].
(b) [6 points] Find P [server is working | first access attempt fails ].
(c) [6 points] Find P [second access attempt fails | first access attempt fails ].
(d) [6 points] Find P [server is working | first and second access attempts fail ].
- [36 points] On the basis of a sensor output X, it is to be decided which hypothesis is true: H 0 or H 1. Suppose that if H 0 is true then X has density f 0 and if H 1 is true then X has density f 1 , where the densities are given by
f 0 (u) =
2 |u| ≤^1 0 |u| > 1 f 1 (u) =
|u| |u| ≤ 1 0 |u| > 1
(a) [6 points] Describe the maximum likelihood (ML) decision rule for deciding which hypothesis is true for observation X.
(b) [6 points] Find the probability of false alarm, pf alse alarm, for the ML rule.
(c) [6 points] Find the probability of missed detection, pmiss, for the ML rule.
- [24 points] Suppose X and Y are jointly Gaussian with parameters μX = μY = 0, σ^2 X = 4, σ Y^2 = 9, and ρ = 0.5. Let W = 2X + 3Y and Z = X + aY , where a is a real valued constant.
(a) [6 points] Express Cov(W, Z) as a simple function of a.
(b) [6 points] For what value(s) of a are W and Z jointly Gaussian?
(c) [6 points] For what value(s) of a are W and Z independent?
(d) [6 points] For what value(s) of a do W and Z fail to be jointly continuous (i.e., fail to have a joint pdf)?
- [56 points] The jointly continuous random variables X and Y have joint pdf given by
fX,Y (u, v) =
2 / 3 , 0 ≤ u ≤ 1 , 0 ≤ v ≤ 1 , 0 ≤ u + v ≤ 1 , 4 / 3 , 0 ≤ u ≤ 1 , 0 ≤ v ≤ 1 , 1 < u + v ≤ 2 , 0 , elsewhere.
(a) [8 points] Find the marginal pdf of X and draw a neat sketch of fX (u).
(b) [8 points] Find the conditional pdf of Y given that X = 2/3 and draw a neat sketch of fY | X (v | 2 /3).
(c) [8 points] Find P {X^2 + Y 2 ≤ 1 }.
(d) [8 points] Find P {X^2 + Y 2 ≤ 1 | (X − 1)^2 + (Y − 1)^2 ≤ 1 }.
- [35 points] Let (X, Y ) be uniformly distributed over the region shown. Also shown, for your convenience, is the pdf of X, which is the same as the pdf of Y.
0.5 1.0 1.
u
v
2/
4/
f (u) X
u 0 0.
(a) [7 points] Find the function g(X) which gives the minimum mean square error (MMSE) estimator of Y given X.
(b) [7 points] Find the (average) mean square error for the MMSE estimator of Y given X.
The remainder of this problem uses the same pdfs:
0.5 1.0 1.
u
v
2/
4/
f (^) X(u)
u 0 0.
(c) [7 points] Find E[X].
(d) [7 points] Find E[XY ].
(e) [7 points] Sketch the linear function L(u), of the form au + b, such that L(X) is the linear estimator of Y based on X with the minimum MSE. (You are not required to compute the exact numerical values, but the smaller the MSE for the linear estimator you sketch, the higher the score.)
