1.) Let X and Y be random variables that take on values x1, x2, &he...

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1.) Let X and Y be random variables that take on values x1, x2, …, xr and y1, y2, …, ys respectively. Let Z = X + Y. (i) Show that H(Z|X) = H(Y|X) (ii) If X and Y are independent, then show how H(Y) < = H(Z) and H(X) < = H(Z) (iii)Under what condition will H(Z) = H(X) + H(Y) (v) Calculate the probability that if somebody is “tall” (meaning taller than 6 ft), that person must be male. Assume that the probability of being male is p(M) = 0.5 and so likewise for being female p(F) = 0.5. Suppose that 20% of males are T (i.e. tall): p(T|M) = 0.2; and that 6% of females are tall: p(T|F) = 0.06. Find p(M|T). If you know that somebody is male, how much information do you gain (in bits) by learning that he is also tall? How much do you gain by learning that a female is tall? Finally, how much information do you gain from learning that a tall person is female? 2.)Consider the Binary Symmetric Channel (BSC) . It is a channel that transports 1’s and 0’s from the transmitter (Tx) to the receiver (Ry). It makes an error occasionally, with probability p. A BSC flips a 1 to 0 and vice-versa with equal probability. Let X and Y be binary random variables that represent the input and output of this BSC respectively. Let the input symbols be equally likely and the output symbols depend upon the input according to the channel transition probabilities. i) What is the entropy H(X) of this binary source? ii) What is the conditional entropy H(X|Y) ? iii)Find the values of H(X|Y) for the following values of p: a) 0.5 b) 0.2 c) 0.8 d) 0.1 e)0.9 iv)Find the values of Average Mutual Information I(X;Y) for the following values of p: a) 0.1 b) 0.5 c) 0.9 d) 0.3 e)0.7 v) Fill in the blanks: For this BSC, as we increase the value of p from 0 to 0.5 the value of I(X;Y) (increases/decreases). At p = 0.5 the value of H(X) = and the value of I(X;Y) = . For this BSC as we increase the value of p from 0.5 to 1 the value of I(X;Y) (increases/decreases). At p = 0, the value of H(X) = and the value of I(X;Y) = . At p = 1, the value of H(X) = and the value of I(X;Y) = . vi.) Now, consider a binary symmetric communication channel, whose input source is the alphabet X = {0, 1} with probabilities {0.5, 0.5}; output alphabet Y = { 0, 1}; and with channel matrix: 1 – α α where α is the probability of transmission error. (i ) What is the entropy of the source, H(X)? (ii ) What is the probability distribution of the outputs, p(Y), and what is the entropy of this output distribution, H(Y)? (iii)What is the joint probability distribution for the source and the output, p(X,Y), and what is the joint entropy, H(X; Y )? (iv) What is the mutual information of this channel, I(X; Y )? (v) How many values are there for α for which the mutual information of this channel is maximal and what are those values ?

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