Consider two random variables X and Y defined over {0, 1} with joint probability distribution pXY . Consider a conditional probability distribution pY |X defined by pY |X(0|0) = 1, pY |X(1|0) = 2, pY |X(0|1) = 3, pY |X(1|1) = 4.

1. Justify that pY |X(1|0) = 1 1 and pY |X(1|1) = 1 3. 2. Assume that 1 = 3 = 1/4 and assume that pX(0) = 1/2. Determine the probability distributions pXY and pY . 3. Compute (give first the exact value simplified as much as possible, then give a numerical approximation) H(X), H(Y ), H(XY ), H(X|Y ), and H(Y |X)