Say that we want to send a bit X from a transmitter to a receiver. We model X as Bernoulli(1/2). The issue is that each transmitted bit may be corrupted (i.e., flipped from a 0 to 1 or a 1 to a 0) with probability 1/4, independently of other bits. One way to overcome this noise is to repeat transmissions several times and take a majority vote among the received bits. We assume that the bit is repeated three times, and let Y be the number of 1's observed at the receiver. It follows that Y given X = 0 is Binomial(3, 1/4) and Y given X = 1 is Binomial (3,3/4). (a) Write out the joint PMF Px,y(x, y) as a table. (b) Determine the marginal PMF Py(y). (c) Let A = {2,3} and note that if YE A, then the majority of the three transmissions result in a 1 observed at the receiver. Calculate P[YE A].

Say that we want to send a bit X from a transmitter to a receiver. We model X as Bernoulli(1/2). The issue is that each transmitted bit may be corrupted (i.e., flipped from a 0 to 1 or a 1 to a 0) with probability 1/4, independently of other bits. One way to overcome this noise is to repeat transmissions several times and take a majority vote among the received bits. We assume that the bit is repeated three times, and let Y be the number of 1's observed at the receiver. It follows that Y given X = 0 is Binomial(3, 1/4) and Y given X = 1 is Binomial (3,3/4). (a) Write out the joint PMF Px,y(x, y) as a table. (b) Determine the marginal PMF Py(y). (c) Let A = {2,3} and note that if YE A, then the majority of the three transmissions result in a 1 observed at the receiver. Calculate P[YE A].

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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