Chapter Review 547
(e) Note that QŽ is the disjoint union of the events E* and E*. Use this observation
and the Theorem of Total Probability to give an expression for P(W).
(f) Give an expression in terms of P2 and P4 for the probability that nodes n2 and n4
both fail. Justify your answer.
(g) Note that if n, works, then the system works unless n2 and n4 both fail. Use this
observation and the result of part (f) to give an expression for P(W I E*).
(h) Note that if n I fails, then the system can be thought of as a new, simpler sys-
tem consisting of n3 followed by the parallel pair n2, n4. This new system works
provided n3 and at least one of n2, n4 work (compare with Exercise 1). Use this
observation to give an expression for P(WI E*).
(i) Refine the expression for P(W) found in part (e), writing it in terms of the pi.
- Let pi denote the probability that any particular code symbol is erroneously transmit-
ted through a communication system. Assume that on different symbols, errors occur
independently of one another. Suppose also that with probability P2, an erroneous
symbol is corrected on receipt. Let X denote the number of correct symbols in a mes-
sage block consisting of n symbols (count after the correction process has been carried
out). What is the probability distribution of X?
- A computer disk storage device has 10 concentric tracks (numbered 1, 2, ..., 10 from
outermost to innermost) and a single access arm. Let p, be the probability that any
particular request for data will take the arm to track i where 1 < i < 10. Assume that
the tracks accessed in successive seeks form an independent process. Let X be the
number of tracks over which the access arm passes during two successive requests.
Here, if the next track is different from the current track, X counts all the intermediate
tracks plus the new track. For example, going from track 3 to track 7, the arm passes
over tracks 4, 5, and 6 and then lands at track 7, so this gives X = 4. If two succes-
sive requests are for the same track, then X = 0. Hence, the possible values of X are
0, 1..., 9. Compute the probability density function of X. (Hint: P(the arm is now
on track i and X = j) = P(X = j arm now on i) • pi. After writing the conditional
probability in terms of pl, P2,..., pl0 using the Law of Total Probability, the desired
probability is obtained by summing over i).
- An electronic system consisting of n components in series fails if at least one compo-
nent of the system fails. An electronic system consisting of n components in parallel
fails only if all n components fail. Two parallel systems of three elements each are in
series as shown:
System System2
Let Aij be the event that component i in system j fails, where 1 < i < 3 and 1 < j <
- Write the event that the system fails using the terms Aij.
