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Magazine Problem 1
Given a discrete random variable \(X\), define the event \(A_x\) by $$ A_x=\\{s \in S \mid X(s)=x\\} . $$ Show that the family of events \(\left\\{A_x\right\\}\) defines an event space.
Problem 3
A mischievous student wants to break into a computer file, which is password- protected. Assume that there are \(n\) equally likely passwords, and that the student chooses passwords independently and at random and tries them. Let \(N_n\) be the number of trials required to break into the file. Determine the pmf of \(N_n\) (a) if unsuccessful passwords are not eliminated from further selections, and (b) if they are.
Problem 8
The probability of error in the transmission of a bit over a communication channel is \(p=10^{-4}\). What is the probability of more than three errors in transmitting a block of 1000 bits?
Problem 9
Assume that the number of messages input to a communication channel in an interval of duration \(t\) seconds is Poisson distributed with parameter \(0.3 t\). Compute the probabilities of the following events: (a) Exactly three messages will arrive during a 10 s interval (b) At most 20 messages arrive in a period of \(20 \mathrm{~s}\) (c) The number of message arrivals in an interval of \(5 \mathrm{~s}\) duration is between three and seven.
Problem 10
VLSI chips, essential to the running of a computer system, fail in accordance with a Poisson distribution with the rate of one chip in about 5 weeks. If there are two spare chips on hand, and if a new supply will arrive in 8 weeks, what is the probability that during the next 8 weeks the system will be down for a week or more, owing to a lack of chips?
