You set up an interplanetary communications channel with Mars in hopes of establishing communications with a recently landed rover.

(a) Suppose each message you send on this channel is independently corrupted with probability 0.01. Let N be the random variable that counts the number of transmissions before the first erroneous transmission occurs (to be precise, the last erroneous transmission also counts). What is the PMF of N? What is E[N] and std(N), i.e., the expectation and the standard deviation of the transmissions before the first-error is encountered?

(Note: If you can, with justification, identify this as one of the named random variables from class, you may simply write down these quantities - a first principles derivation is not required.) (b) You have been observing the channel for a whole day and there have been k error-free transmissions thus far. Given that you know this information, compute the probability that the n-th transmission is the first time an error occurs, where n > k. (c) Let M = aN +b, for some real numbers a and b. Write down M() the characteristic function of M in terms of N(), the characteristic function of N. (d) Use the above result to derive E[M] and Var[M] (note: you cannot simply use formulas for the expectation and variance of a linear function of a random variable).