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Chapter 3: Q96E (page 137)

The negative binomial rv \({\bf{X}}\) was defined as the number of \({\bf{F}}\)鈥檚 preceding the rth \({\bf{S}}\). Let \({\bf{Y}}{\rm{ }}{\bf{5}}\) the number of trials necessary to obtain the rth \({\bf{S}}\). In the same manner in which the pmf of \({\bf{X}}\) was derived, derive the pmf of \({\bf{Y}}\)

Short Answer

Expert verified

The pmf of\(Y\)is \(p(y) = \left( {\begin{array}{*{20}{c}}{y - r + r - 1}\\{r - 1}\end{array}} \right){p^r}{(1 - p)^{y - r}},\;\;\;y \in \{ r,r + 1,r + 2, \ldots \} \)

Step by step solution

01

Concept introduction

A binomial is a mathematical expression consisting of two terms connected by a plus sign or minus sign.

02

Determine the pmf of Y

The probability mass function of the negative binomial random variable \(X\) with parameters \(r\)and \(p\), for \(x \in \mathbb{N}\), is if there are \(r\)successes with \(p\)probability of success.

\(nb(x;r,p) = \left( {\begin{array}{*{20}{c}}{x + r - 1}\\{r - 1}\end{array}} \right){p^r}{(1 - p)^x}\)

Define \(Y\)as the described random variable.

\({S_y} = \left\{ {{\bf{y}}{\rm{ trials necessary to obtain }}{r^{th}}{\rm{ success }}} \right\}\)

\({F_y} = \left\{ {{\bf{y}}{\rm{ failures preceding the }}{{\bf{r}}^{{\rm{th }}}}{\rm{ success }}} \right\}\)

\(p(y) = P(Y = y) = P\left( {{S_y}} \right)\mathop = \limits^{(1)} P\left( {{F_{y - r}}} \right)\)

\( = nb(y - r;r,p) = \left( {\begin{array}{*{20}{c}}{y - r + r - 1}\\{r - 1}\end{array}} \right){p^r}{(1 - p)^{y - r}},\;\;\;y \in \{ r,r + 1,r + 2, \ldots \} \)

If there are total of \(y\)trials necessary to obtain \({r^{{\rm{th }}}}\)success, then there must be \(y - r\)failures preceding \({r^{{\rm{th }}}}\)success.

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Most popular questions from this chapter

Organisms are present in ballast water discharged from a ship according to a Poisson process with a concentration of\({\bf{10}}\)organisms/m3 (the article 鈥淐ounting at Low Concentrations: The Statistical Challenges of Verifying Ballast Water Discharge Standards鈥� (Ecological Applications) considers using the Poisson process for this purpose). a. What is the probability that one cubic meter of discharge contains at least 8 organisms? b. What is the probability that the number of organisms in\({\bf{1}}.{\bf{5}}{\rm{ }}{{\bf{m}}^3}\)of discharge exceeds its mean value by more than one standard deviation? c. For what amount of discharge would the probability of containing at least one organism be\(.{\bf{999}}\)?

A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table.

X

0

1

2

3

4

5

6

p(x)

.10

.15

.20

.25

.20

.06

.04

Calculate the probability of each of the following events.

a. {at most three lines are in use}

b. {fewer than three lines are in use}

c. {at least three lines are in use}

d. {between two and five lines, inclusive, are in use}

e. {between two and four lines, inclusive, are not in use}

f. {at least four lines are not in use}

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 \({\rm{10}}\) times. Assuming that the results of successive transmissions are independent of one another and that the probability of any particular transmission being successful is \({\rm{p}}\), determine the probability mass function of the rv \({\rm{X = }}\)the number of times a packet is transmitted. Then obtain an expression for the expected number of times a packet is transmitted.

A new battery鈥檚 voltage may be acceptable (A) or unacceptable (U). A certain flashlight requires two batteries, so batteries will be independently selected and tested until two acceptable ones have been found. Suppose that 90% of all batteries have acceptable voltages. Let Y denote the number of batteries that must be tested.

a. What is\(p\left( 2 \right)\), that is, \(P\left( {Y = 2} \right)\),?

b. What is\(p\left( 2 \right)\)? (Hint: There are two different outcomes that result in\(Y = 3\).)

c. To have \(Y = 5\), what must be true of the fifth battery selected? List the four outcomes for which Y = 5 and then determine\(p\left( 5 \right)\).

d. Use the pattern in your answers for parts (a)鈥�(c) to obtain a general formula \(p\left( y \right)\).

Each of \({\rm{12}}\)refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that \({\rm{7}}\) of these refrigerators have a defective compressor and the other \({\rm{5}}\) have less serious problems. If the refrigerators are examined in random order, let\({\rm{X}}\)be the number among the first \({\rm{6}}\) examined that have a defective compressor.

a. Calculate\({\rm{P(X = 4)}}\)and \(P(X拢 4)\)

b. Determine the probability that \({\rm{X}}\) exceeds its mean value by more than \({\rm{1}}\) standard deviation.

c. Consider a large shipment of \({\rm{400}}\)\({\rm{40}}\) refrigerators, of which 40 have defective compressors. If \({\rm{X}}\) is the number among \({\rm{15}}\) randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately) P(X拢5)than to use the hypergeometric \({\rm{pmf}}\).
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