below is defined by the transition probabilities: P[Y = 0|X = 0] = 1, P[Y = 1|X = 1] = 1 - €, and P[Y = 0|X = 1] = e, where random variable X is the channel input bit, and Y is the received bit. Let the data source produce "on" bits with probability P[X = 1] = When using an n-fold repetition code (recall the lecture discussion), a bit generated by the source is transmitted n times (independent transmissions) through the channel. At the receiver, the decoder accumulates the sequence of n received bits and then estimates the source bit. (Answers below should only be in terms of n, p and e.) = p. a) First, establish for a single bit transmission the probability of success, P[Y = X] b) For the rest of the problem, the n-fold repetition code is used. Suppose the receiver receives a sequence of n zeros. What is the probability that the source bit was indeed zero? c) What is the probability that the source bit was zero, if the receiver accumulates a sequence that contains n – 1 zeros and a single one? d) In general, assuming certainty what the source bit was? > 0, what is the probability that the decoder will know with 1 Y 1 + 1 1 - €
below is defined by the transition probabilities: P[Y = 0|X = 0] = 1, P[Y = 1|X = 1] = 1 - €, and P[Y = 0|X = 1] = e, where random variable X is the channel input bit, and Y is the received bit. Let the data source produce "on" bits with probability P[X = 1] = When using an n-fold repetition code (recall the lecture discussion), a bit generated by the source is transmitted n times (independent transmissions) through the channel. At the receiver, the decoder accumulates the sequence of n received bits and then estimates the source bit. (Answers below should only be in terms of n, p and e.) = p. a) First, establish for a single bit transmission the probability of success, P[Y = X] b) For the rest of the problem, the n-fold repetition code is used. Suppose the receiver receives a sequence of n zeros. What is the probability that the source bit was indeed zero? c) What is the probability that the source bit was zero, if the receiver accumulates a sequence that contains n – 1 zeros and a single one? d) In general, assuming certainty what the source bit was? > 0, what is the probability that the decoder will know with 1 Y 1 + 1 1 - €
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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On asymmetric binary channels: The binary “Z” channel shown below is
![below is defined by the transition probabilities: P[Y = 0|X = 0] = 1, P[Y = 1|X = 1]
1– €, and P[Y = 0|X = 1] = €, where random variable X is the channel input bit, and Y
is the received bit. Let the data source produce “on" bits with probability P[X = 1]
When using an n-fold repetition code (recall the lecture discussion), a bit generated by
the source is transmitted n times (independent transmissions) through the channel. At the
receiver, the decoder accumulates the sequence of n received bits and then estimates the
source bit. (Answers below should only be in terms of n, p and e.)
р.
a) First, establish for a single bit transmission the probability of success, P[Y = X]
b) For the rest of the problem, the n-fold repetition code is used. Suppose the receiver
receives a sequence of n zeros. What is the probability that the source bit was indeed
zero?
c) What is the probability that the source bit was zero, if the receiver accumulates a
sequence that contains n – 1 zeros and a single one?
d) In general, assuming e > 0, what is the probability that the decoder will know with
certainty what the source bit was?
1
X
Y
1
1
1- €](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F099a9854-7cc3-46b9-a37c-ec0486b4a12a%2F2fd6261c-fccc-42e8-9eeb-2c830ca135d0%2Fjri8fk_processed.png&w=3840&q=75)
Transcribed Image Text:below is defined by the transition probabilities: P[Y = 0|X = 0] = 1, P[Y = 1|X = 1]
1– €, and P[Y = 0|X = 1] = €, where random variable X is the channel input bit, and Y
is the received bit. Let the data source produce “on" bits with probability P[X = 1]
When using an n-fold repetition code (recall the lecture discussion), a bit generated by
the source is transmitted n times (independent transmissions) through the channel. At the
receiver, the decoder accumulates the sequence of n received bits and then estimates the
source bit. (Answers below should only be in terms of n, p and e.)
р.
a) First, establish for a single bit transmission the probability of success, P[Y = X]
b) For the rest of the problem, the n-fold repetition code is used. Suppose the receiver
receives a sequence of n zeros. What is the probability that the source bit was indeed
zero?
c) What is the probability that the source bit was zero, if the receiver accumulates a
sequence that contains n – 1 zeros and a single one?
d) In general, assuming e > 0, what is the probability that the decoder will know with
certainty what the source bit was?
1
X
Y
1
1
1- €
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