When a two-way paging system transmits a message, the probability that the messa

Question

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 acknowledgment signal (ACK) to the paging system. If the paging system does not receive the ACK, it sends the message again.

(a)   Now assume that p=0.65, and define N to be the number of times that the message was sent by the pager.
Then N is a (_________) (_____(___ ) random variable. That is, identify the type of random variable and its parameter(s).

(b)   Find the PMF of N.

Explanation / Answer

a) p = 0.65 , N = number of times the message has to be sent by the pager.
N follows Geometric (p) where p = 0.65

b)   N = 1 w.p p ... it recieves in the first attempt
2 w.p p*(1-p)
3 w.p p*(1-p)^2
.......... inf

c) paging company has the goal of P (N <= 4) = 0.99
p + p(1-p) + ... p*(1-p)^3 = p*(1- (1-p)^4)/p = 1 - (1 - p)^4 = 0.99
==> (1-p)^4 = 0.01 ==> 1-p = 0.316 ==> p = 0.684
so the value of p > 0.684 achieves the goal

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