Chapter 3, Problem 122SE

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9th Edition

Jay L. Devore

ISBN: 9781305251809

Textbook Problem

Consider a communication source that transmits packets containing digitized speech. After each transmission, the receiver sends a message indicating whether the transmission was successful or unsuccessful. If a transmission is unsuccessful, the packet is re-sent. Suppose a voice packet can be transmitted a maximum of 10 times. Assuming that the results of successive transmissions are independent of one another and that the probability of any particular transmission being successful is *p*, determine the probability mass function of the rv *X* = the number of times a packet is transmitted. Then obtain an expression for the expected number of times a packet is transmitted.

To determine

Find the pmf of *X*.

Find the expression for the expected number of times a packet is transmitted.

Explanation

**Given info:**

A communication source transmits packets containing digitized speech. The receivers sends a message whether the transmission was successful or not after each transmission. If it is unsuccessful, the packet is re-sent. A packet can be re transmitted maximum 10 times.

*X* be the number of times a packet is transmitted.

**Calculation:**

*The pmf of X:*

The all possible values of *x* are 1,2ā¦10.

The probability of getting x times transmitted is 1^{st} (*x* ā 1) times failed, at xth time it was completed.

Thus,

If *x* = 10, then

Hence, the pmf is

Sect-3.1 P-1ESect-3.1 P-3ESect-3.1 P-4ESect-3.1 P-5ESect-3.1 P-6ESect-3.1 P-7ESect-3.1 P-8ESect-3.1 P-9ESect-3.1 P-10ESect-3.2 P-11E

Sect-3.2 P-12ESect-3.2 P-13ESect-3.2 P-14ESect-3.2 P-15ESect-3.2 P-16ESect-3.2 P-17ESect-3.2 P-18ESect-3.2 P-19ESect-3.2 P-20ESect-3.2 P-21ESect-3.2 P-22ESect-3.2 P-23ESect-3.2 P-24ESect-3.2 P-25ESect-3.2 P-26ESect-3.2 P-27ESect-3.2 P-28ESect-3.3 P-29ESect-3.3 P-30ESect-3.3 P-31ESect-3.3 P-32ESect-3.3 P-33ESect-3.3 P-34ESect-3.3 P-35ESect-3.3 P-36ESect-3.3 P-37ESect-3.3 P-38ESect-3.3 P-39ESect-3.3 P-40ESect-3.3 P-41ESect-3.3 P-42ESect-3.3 P-43ESect-3.3 P-44ESect-3.3 P-45ESect-3.4 P-46ESect-3.4 P-47ESect-3.4 P-48ESect-3.4 P-49ESect-3.4 P-50ESect-3.4 P-51ESect-3.4 P-52ESect-3.4 P-53ESect-3.4 P-54ESect-3.4 P-55ESect-3.4 P-56ESect-3.4 P-57ESect-3.4 P-58ESect-3.4 P-59ESect-3.4 P-60ESect-3.4 P-61ESect-3.4 P-62ESect-3.4 P-63ESect-3.4 P-64ESect-3.4 P-65ESect-3.4 P-66ESect-3.4 P-67ESect-3.5 P-68ESect-3.5 P-69ESect-3.5 P-70ESect-3.5 P-71ESect-3.5 P-72ESect-3.5 P-73ESect-3.5 P-74ESect-3.5 P-75ESect-3.5 P-76ESect-3.5 P-77ESect-3.5 P-78ESect-3.6 P-79ESect-3.6 P-80ESect-3.6 P-81ESect-3.6 P-82ESect-3.6 P-83ESect-3.6 P-84ESect-3.6 P-85ESect-3.6 P-86ESect-3.6 P-87ESect-3.6 P-88ESect-3.6 P-89ESect-3.6 P-90ESect-3.6 P-91ESect-3.6 P-92ESect-3.6 P-93ECh-3 P-94SECh-3 P-95SECh-3 P-96SECh-3 P-97SECh-3 P-98SECh-3 P-99SECh-3 P-100SECh-3 P-101SECh-3 P-102SECh-3 P-103SECh-3 P-104SECh-3 P-105SECh-3 P-106SECh-3 P-107SECh-3 P-108SECh-3 P-109SECh-3 P-110SECh-3 P-111SECh-3 P-112SECh-3 P-113SECh-3 P-114SECh-3 P-115SECh-3 P-116SECh-3 P-117SECh-3 P-118SECh-3 P-119SECh-3 P-120SECh-3 P-121SECh-3 P-122SE

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