1. In a transmission system a 0 is encodded as 00000 and 1 as 11111 and these bits are sent through the binary symmetric channel where the bit error probality is p. At the receiving end the decoding...

1. In a transmission system a 0 is encodded as 00000 and 1 as 11111 and these bits are sent through the binary symmetric channel where the bit error probality is p. At the receiving end the decoding is done by majority voting. What is the probability of error PE assuming p = 0.1? When 0 is encoded as 0000000 and 1 as 1111111 and the decoding is done again by majority voting, what is the value of PE for p = 0.1?2. The alpha fetal protein test is meant to detect spina bifida in unborn babies, a condition that affects 1 out of 1000 children who are born. Let B be the event that the baby has spina bifida and Bc be the event that it does not. The literature on the test indicates that 5% of the time a healthy baby will cause a positive reaction. We will assume that the test is positive 100% of the time spina bifida is present. Your doctor has just told you that your alpha fetal protein test was positive. What is the probability that your baby has spina bifida ?3. A random number is selected uniformly from 0; 1; 2; 3; 4; 5; 6; 7 without replacement until 3 is chosen. Let X denote the number of selection. Find the entropy H(X) in bits.4. A box of 50 semiconductor chips includes 3 defective ones. 5 chips are randomly chosen from this box. Let X denote the number of defective chips. Find the entropy of H(X) in bits.5. World Series. The world series is a seven game-series that terminates as soon as either team wins four games. Let X be the random variable that represents the outcome of a World Series between teams A and B; some possible values of X are AAAA,ABABBB, and BABABAA. Let Y be the number of games played, which ranges from 4 to 7. Assuming that A and B are equally matched and the games are independent, calculate H(X) and H(Y ).6. Erasure Channel. Consider the discrete memoryless channel as shown in Figure 1. Assuming P(X = 0) = 2 3 and P(X = 1) = 1 3, and p = 1 4, findX10 01e1 − ppp1 − pYFigure 1: Erasure Channel(a) H(X),H(Y ) (b) H(Y/X),H(X/Y ) (c) H(X,Y ) (d) I(X,Y )7. A fair coin is flipped until the first head occurs. Let X denote the number of flips required. Find H(X).