A 4 - ary digital communication system has an input X and output Y. Both X and Y can only be one of the four so - called letters 1, 2, 3, and 4, which is why the system is called 4 - ary. The channel between X and Y is specified by a matrix whose (i, j)th entry is Py|x(J|i), i, j = 1>2, 3, 4. The matrix is as follows: |0. 1 0. 2 0. 3 0. 4 0. 2 0. 1 0. 4 0. 3 0. 4 0. 3 0. 2 0. 1 0. 3 0. 2 0. 4 0. 1 | Suppose that all the 4 input letters, 1, 2, 3, 4, are used with equal probability: Px(i) = 1/4, i=1, 2, 3, 4. Find the joint PMF of X and Y. Find the marginal PMF of Y. Given that Y = 2, what is the distribution of X? Compute E[Y] and E[X|Y = 2]. Given that Y = 2, which input letter is most likely the transmitted letter? If you decide on this letter as the estimated input letter, what is the probability that this decision is wrong?
Show transcribed image text A 4 - ary digital communication system has an input X and output Y. Both X and Y can only be one of the four so - called letters 1, 2, 3, and 4, which is why the system is called 4 - ary. The channel between X and Y is specified by a matrix whose (i, j)th entry is Py|x(J|i), i, j = 1>2, 3, 4. The matrix is as follows: |0. 1 0. 2 0. 3 0. 4 0. 2 0. 1 0. 4 0. 3 0. 4 0. 3 0. 2 0. 1 0. 3 0. 2 0. 4 0. 1 | Suppose that all the 4 input letters, 1, 2, 3, 4, are used with equal probability: Px(i) = 1/4, i=1, 2, 3, 4. Find the joint PMF of X and Y. Find the marginal PMF of Y. Given that Y = 2, what is the distribution of X? Compute E[Y] and E[X|Y = 2]. Given that Y = 2, which input letter is most likely the transmitted letter? If you decide on this letter as the estimated input letter, what is the probability that this decision is wrong?