Refer back to the communication system of Example 3.51. Suppose a voice packet can be transmitted a maximum of 10 times; i.e., if the 10th attempt fails, no 11th attempt is made to re-transmit the voice packet. Let X = the number of times a message is transmitted. Assuming each transmission succeeds with probability p, determine the pmf of X. Then obtain an expression for the expected number of times a packet is transmitted.

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Example 3.51 In many communication systems, a receiver will send a short signal back to the transmitter to indicate whether a message has been received correctly or with errors. (These signals are often called an acknowledgement and a nonacknowledgement, respectively. Bit sum checks and other tools are used by the receiver to determine the absence or presence of errors.) Assume we are using such a system in a noisy channel, so that each message is sent error-free with probability .86, independent of all other messages. What is the probability that in 10 transmissions, exactly 8 will succeed? What is the probability the system will require exactly 10 attempts to successfully transmit 8 messages? While these two questions may sound similar, they require two different models for solution. To answer the first question, let X = the number of successful transmissions among the 10. Then X~ Bin(10, .86), and the answer is (86)*(.14)² = 2639 P(X= 8) = b(8; 10, .86) = 10 8