Problem 4) When a conventional paging system transmits a message, the probabilit

Question

Problem 4) When a conventional paging system transmits a message, the probability that the message will be received by the pager it is sent to is p. To be confident that a message is received at least once, a system transmits the message n times. a) Assuming al transmissions are independent, what is the PMF of K, the number of times the pager receives the same message. b) Assume p = 0.8. What is the minimum value of n that produces a probability of 0.95 of receiving the message at least once? Problem 5) When a two-way paging system transmits a message, the probability that the message will be received by the pager it is sent to is p. When the pager receives the message, it transmits an acknowledgement signal (ACK) to the paging system. If the paging system does not receive the ACK, it sends the message again. a) What is the PMF of N, the number of times the system sends the same message? b) The paging company wants to limit the number of times it has to send the same message. It has a goal of PIN s 3120.95. What is the minimum value of p necessary to achieve the goal? Problem 6) The number of bits B in a fax transmission is a geometric (p 2.5 10) random variable. What is the probability P[B> 500,0001 that a fax has over 500,000 bits?

Explanation / Answer

Problem 4

Part (a)

K ~ B(n, p) and hence pmf of K is :

P(K = k) = p(k) = (nCk)pk(1 - p)n – k ANSWER

Part (b)

We want the minimum value of n at p = 0.8 which produces a probability 0.95 of receiving the message at least once, i.e., P(K 1) which in turn is equal to 1 – P(K = 0)

= 1 – 0.2n

This must be 0.95. Or

1 – 0.2n > 0.95 or

0.2n < 0.05

Since it is easy to visualize that 0.22 = 0.04, which is less than 0.05, minimum value of n is 2.

ANSWER

[Analytically, nlog0.2 < log 0.05 and solve the inequality for n]

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