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Problem Set 5 for ECE 413 at University of Illinois, Spring 2005, Assignments of Statistics

University of Illinois - Urbana-ChampaignStatistics

Problem set 5 for the ece 413 course at the university of illinois, spring 2005. The problem set includes six problems covering topics such as probability theory, conditional probability, and discrete random variables. Students are expected to solve these problems and submit them by the given deadline. The document also includes reminders about upcoming exams and required readings.

Typology: Assignments

Pre 2010

Uploaded on 02/24/2010

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University of Illinois Spring 2005
ECE 413: Problem Set 5
Due: Wednesday February 23 at the beginning of class.
Reminder: Hour Exam I is on Monday February 28, 7 p.m. in 269 Everitt Lab.
Reading: Ross, Chapter 4, Chapter 3
Noncredit Exercises: DO NOT turn these in.
Chapter 3: Problems 1, 2, 5, 10, 12, 16, 31, 38, 39, 44
Theoretical Exercises 1, 2, 8; Self-Test Problems 1-10.
This Problem Set contains six problems
1.[“I am from Iowa; I only work in outer space .. .”] Each box of Cornies, the breakfast of
silver medalists, contains either a picture of Luke Skywalker or a picture of Darth Vader
with probabilities 2
3and 1
3respectively. The contents of each box may be considered to
be independent of the contents of other boxes. Little Jimmy T. Kirk of Cedar Rapids,
Iowa, asks his mother to buy boxes of Cornies until he has accumulated at least one
picture of both Luke and Darth.
(a) What is the minimum number of boxes that Mrs Kirk must purchase?
(b) Let Xdenote the number of boxes that Mrs Kirk buys till Jimmy has his heart’s
desire. What is the pmf of X? What is the expected value of X?
(c) The following year, pictures of Homer and Bart replace those of Luke and Darth
in boxes of Cornies. Being even more spoiled than before, Jimmy wants to have
at least two pictures of each. Repeat parts (a) and (b) for these conditions.
2. A long message is divided into Lpackets of Nbits each (including headers, addresses,
timestamps, data bits, CRC bits, tail, flags etc.) and transmitted over a channel with
bit error probability p. If the CRC detects that a packet is received in error, the packet
transmission is repeated. But, if a packet has been transmitted 5 times and still has
not been received correctly on the fifth try, then it is deemed to be lost and is not
transmitted again.
(a) What is the probability that the CRC indicates no error in a received packet?
(b) What is the probability that a packet is transmitted successfully (i.e. is not
deemed to be lost)?
(c) Let Xidenote the number of times that the i-th packet is transmiitted. What is
the pmf of Xi? What is E[Xi]?
(d) What is the probability that none of the Lpackets are lost?
3. This problem on conditional probability has three unrelated parts:
(a) If P(A|B) = 0.3, P (Ac|Bc) = 0.4, and P(B) = 0.7, find P(A|Bc), P (A), and
P(B|A).
(b) If P(E) = 1
4, P (F|E) = 1
2, and P(E|F) = 1
3, find P(F).
(c) If P(G) = P(H) = 2
3, show that P(G|H)≥1
2.
pf2

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Download Problem Set 5 for ECE 413 at University of Illinois, Spring 2005 and more Assignments Statistics in PDF only on Docsity!

University of Illinois Spring 2005

ECE 413: Problem Set 5

Due: Wednesday February 23 at the beginning of class. Reminder: Hour Exam I is on Monday February 28, 7 p.m. in 269 Everitt Lab. Reading: Ross, Chapter 4, Chapter 3 Noncredit Exercises: DO NOT turn these in. Chapter 3: Problems 1, 2, 5, 10, 12, 16, 31, 38, 39, 44 Theoretical Exercises 1, 2, 8; Self-Test Problems 1-10.

This Problem Set contains six problems

  1. [“I am from Iowa; I only work in outer space... ”] Each box of Cornies, the breakfast of silver medalists, contains either a picture of Luke Skywalker or a picture of Darth Vader with probabilities 23 and 13 respectively. The contents of each box may be considered to be independent of the contents of other boxes. Little Jimmy T. Kirk of Cedar Rapids, Iowa, asks his mother to buy boxes of Cornies until he has accumulated at least one picture of both Luke and Darth.

(a) What is the minimum number of boxes that Mrs Kirk must purchase? (b) Let X denote the number of boxes that Mrs Kirk buys till Jimmy has his heart’s desire. What is the pmf of X? What is the expected value of X? (c) The following year, pictures of Homer and Bart replace those of Luke and Darth in boxes of Cornies. Being even more spoiled than before, Jimmy wants to have at least two pictures of each. Repeat parts (a) and (b) for these conditions.

  1. A long message is divided into L packets of N bits each (including headers, addresses, timestamps, data bits, CRC bits, tail, flags etc.) and transmitted over a channel with bit error probability p. If the CRC detects that a packet is received in error, the packet transmission is repeated. But, if a packet has been transmitted 5 times and still has not been received correctly on the fifth try, then it is deemed to be lost and is not transmitted again.

(a) What is the probability that the CRC indicates no error in a received packet? (b) What is the probability that a packet is transmitted successfully (i.e. is not deemed to be lost)? (c) Let Xi denote the number of times that the i-th packet is transmiitted. What is the pmf of Xi? What is E[Xi]? (d) What is the probability that none of the L packets are lost?

  1. This problem on conditional probability has three unrelated parts:

(a) If P (A|B) = 0. 3 , P (Ac|Bc) = 0.4, and P (B) = 0.7, find P (A|Bc), P (A), and P (B|A). (b) If P (E) = 14 , P (F |E) = 12 , and P (E|F ) = 13 , find P (F ).

(c) If P (G) = P (H) = 23 , show that P (G|H) ≥ 12.

  1. Let X and Y denote two discrete random variables taking on values 1, 2 , 3. X denotes a number that we wish to transmit over a channel using one of the three signals s 1 , s 2 and s 3. Thus, the signal sX is transmitted. Noise in the channel can corrupt the signal, and thus it is possible that the received signal sY is not the same as the transmitted signal sX. In particular, the transition matrix below gives the (conditional) probability that the receiver hears Y when the transmitter says X.

Transmitted Received Y X 1 2 3 1 0. 8 0. 1 0. 1 2 0. 05 0. 9 0. 05 3 0. 15 0. 05 0. 8

This table is saying, for example, that a transmitted 1 is received as a 1, or 2 or 3 with probabilities 0. 8 , 0 .1, and 0.1 respectively.

(a) Suppose that X has pmf pX (1) = 0.5, pX (2) = 0.25, pX (3) = 0.25. What is the pmf of Y? (b) Given that the receiver heard Y = 3, what are the conditional probabilities of {X = 1}? {X = 2}? {X = 3}?

  1. An urn contains r red and g green balls. Two balls are drawn at random from the urn, with the first ball being returned to the urn (which is then well shaken) before the second ball is drawn. Let R 1 and R 2 respectively denote the events that the first and second balls are red.

(a) What are P (R 1 ) and P (R 2 )? (b) Now suppose that when the first ball is returned to the urn, c additional balls of the same color are also put into the urn (which is then well shaken before the second ball is drawn.) Clearly P (R 1 ) is the same as before, but what is P (R 2 ) now? Remember that the urn now contains r + g + c balls. Simplify your answer and compare to the value of P (R 2 ) that you obtained in part (a). (c) For the experiment of part (b), what is the conditional probability that the urn contained r + c red balls given that R 2 occurred?

  1. We consider Problem 4 of the previous problem set under the condition that 15 of the 105 passengers are arriving in Chicago on a connecting flight. If the connecting flight is on time, all show up at the gate for the flight to Champaign: else none of the 15 make it. The remaining 90 passengers make up their minds individually as before (and independently of the fate of the incoming flight). The number of these who show up at the gate is thus a binomial random variable Z with parameters (90, 0 .9). As before, let X denote the number of passengers presenting themselves at the gate. Let T denote the event that the connecting flight is on time, and A = {X ≤ 100 } the event that all who show up at the gate get to board the flight.

(a) What are the conditional probabilities P {A|T } and P {A|T c}? (b) Suppose that P (T ) = 23. Find P (A). (c) Given that all who showed up got a seat, what is the conditional probability that the connecting flight was on time?