Examfinalsol

University of Illinois ECE313: Final Exam Friday, August 4, 2017, 8-10 a.m. ECEB 3017 o B - Name: (in BLOCK CAPITALS) 20 ( ,"\ VO Summer 2017 NetlD: Signature: Instructions This exam is closed book and closed notes except that two 8.5" x11" sheets of notes is permitted: both sides may be used. No electronic equipment (cell phones, etc.) allowed. Grading The exam consists of 7 problems worth 1. 14 points a total of 100 points. The problems are not weighted equally, so it is best for 2. 10 points you to pace yourself accordingly. Write your answers in the spaces provided, and reduce common fractions to low- 3. 14 points est terms, but DO NOT convert them to decimal fractions (for example, write 4. 14 points 4 3 instead of 24 or 0.75). 4 32 5. 25 points SHOW YOUR WORK; BOX YOUR AN- . SWERS. Answers without appropri- 6. 13 points ate justification will receive very little credit. If you need extra space, use 7. 10 points the back of the previous page. Draw ' a small box around each of your final | Total (100 points) numerical answers.

1. [14 points] A message source that produces a sequence of 4 bits of information is equally likely to be in one of three states: 5, Si, and Ss;. In.state Sk, the 4 bit sequernce can have at most £ 1's, with all such 4 bit sequences being equally likely. For example, in state Sy, the source can produce the sequences 0000, 1000, 0100, 0010, 0001 with each of sequences being produced with probability % Let A be event that the sequence 1010 was produced by the source. (a) [4 points] Find P(AISQ). | 5, 9 ot mosd+ 2 1% ooT o b U D sel= ()0 ) « (D) veude= i 2 'T R | LG I A & Vs = p (Alsy)= 1 "3ng cedvced - ISL) 59-""WL€ §/7 o.ce 5{@ 52.. (b) [6 points] Find P(A). E | ', U?m? | Lo o P"GEQ{@,'!%Y 1;})215;]%(\@:!1%&11.? (7(/3(\: P\{/ALEE> (7/§,) ,_,', {i/(f'fi/!/g{fi @[5;} + V(A/gl) P(SJ) | ~ O - (ll(' | - 1 | | By | S L) ) < ze ) U] VA= 0lals,) ol | | yq ¢ ff |

2. [10 points] Consider a binary hypotheses testing problem with observation X. Under Hy, X ~ Binomial (72, %), while under Hy, X ~ Binomial (72, %) . It is known that (a) [6 points] Determine the MAP decision rule. Express the rule in terms of k as simply as possible. In case of a tie in likelihoods, declare H; to be the hypothesis. ,, | ( vy f ok -4 20 z R A A 7 ¢ - - . £z 3¢ L (b) [4 points] Express the approximate value of psaise aiarm for the MAP rule in terms of the @ function, where Q(c) = [° . :/—2—; exp(—u?/2)du. (To be definite, don't use the continuity correction.) | P = vldeclore il g