uppose your fax machine is transmitting a series of independent codewords. Each codeword consists of three bits, such as "100" or "010". Each bit is independent of others and equally likely to be "1" or "0". The weight of a codeword is found by summing the bits, for example the weight of "111" is 3. (a) What is the probability that a codeword with weight at least 2 occurs before a codeword with weight 0? (b) Given that a codeword of weight 0 was just sent, what is the probability that the next two codewords also have this weight? (c) Whenever "000" or "111" is sent, this will be regarded as a "timing pulse". i. Find the PMF of K, the number of codewords up to, but not including, the third timing pulse. ii. Find the expectation of L, the number of timing pulses in the first 100 codewords. iii. Find the expectation and variance of M , the number of "0"s that have been trans- mitted before the first timing pulse. iv. Find the PMF of N , the number of codewords of weight 3 that occur in 100 code- words. v. Find the conditional PMF of N given L = l

(d) My phone is also sending a sequence of codewords, but it employs a different coding scheme. It starts by sending 4-bit codewords. Whenever a "0000" or "1111" occurs, it switches to sending 3-bit codewords, until a "000" or "111" occurs, at which point it switches to 2-bit codewords. When "00" or "11" occurs, the transmission stops. Find E(Q), where Q is the number of codewords sent by my phone