/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q8E Each time a component is tested,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Q8E (page 99)

Each time a component is tested, the trial is a success (S) or failure (F). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let Y denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of Y, and state which Y value is associated with each one.

Short Answer

Expert verified

Y

Outcome

3

\(\left\{ {SSS} \right\}\)

4

\(\left\{ {FSSS} \right\}\)

5

\(\left\{ {FFSSS,SFSSS} \right\}\)

6

\(\left\{ {FFFSSS,SSFSSS,FSFSSS,SFFSSS} \right\}\)

7

\(\left\{ \begin{array}{l}FFFFSSS,FSSFSSS,FFSFSSS,FSFFSSS,\\SFFFSSS,SSFFSSS,SFSFSSS\end{array} \right\}\)

Step by step solution

01

Given information

The random variable Y is defined as the variable that represents the number of trials necessary to achieve the trial鈥檚 success.

The experiment terminates as soon as a success occurs on three consecutive trials.

02

Determine the possibleY values

Let the trial鈥檚 success is denoted by S and the trial鈥檚 failure is denoted by F.

The random variableY can only take on positive values:\(3,4,5,6,7 \ldots \)as Y counts the three-consecutive success in number of trials.

Therefore, the rv is defined by:

\(\left( {Y:Y \in \left\{ {3,4,5,6, \ldots } \right\}} \right)\)

The random variable Y takes infinite set of values in this experiment.

03

List five possible outcomes with their associated values of Y

Here, Y is defined as the variable of the number of trials required to achieve success and it terminated when success occurs on three consecutive trials.

The outcome of three consecutive successes occurs in at least three trials is\(\left\{ {SSS} \right\}\).

Similarly, the outcome of three consecutive success occur in 4 trials is\(\left\{ {FSSS} \right\}\).

Similarly, the outcome of three consecutive success occur in 5 trials are\(\left\{ {FFSSS,SFSSS} \right\}\).

The following table represent the possible outcomes and their associated values of Y.

Y

Outcome

3

\(\left\{ {SSS} \right\}\)

4

\(\left\{ {FSSS} \right\}\)

5

\(\left\{ {FFSSS,SFSSS} \right\}\)

6

\(\left\{ {FFFSSS,SSFSSS,FSFSSS,SFFSSS} \right\}\)

7

\(\left\{ \begin{array}{l}FFFFSSS,FSSFSSS,FFSFSSS,FSFFSSS,\\SFFFSSS,SSFFSSS,SFSFSSS\end{array} \right\}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In proof testing of circuit boards, the probability that any particular diode will fail is\(.{\bf{01}}\). Suppose a circuit board contains\({\bf{200}}\)diodes. a. How many diodes would you expect to fail, and what is the standard deviation of the number that is expected to fail? b. What is the (approximate) probability that at least four diodes will fail on a randomly selected board? c. If five boards are shipped to a particular customer, how likely is it that at least four of them will work properly? (Aboard works properly only if all its diodes work.)

A newsstand has ordered five copies of a certain issue of a photography magazine. Let \({\rm{X = }}\)the number of individuals who come in to purchase this magazine. If \({\rm{X}}\) has a Poisson distribution with parameter \({\rm{\mu = 4}}\), what is the expected number of copies that are sold?

The mode of a discrete random variable \({\rm{X}}\) with \({\rm{pmf}}\) \({\rm{p(x)}}\) is that value \({{\rm{x}}^{\rm{*}}}\) for which \({\rm{p(x)}}\) is largest (the most probable \({\rm{x}}\) value).

a. Let \({\rm{X}} \sim {\rm{Bin(n,p)}}\). By considering the ratio \({\rm{b(x + 1;n,p)/b(x;n,p)}}\), show that \({\rm{b(x;n,p)}}\) increases with x as long as \({\rm{x < np - (1 - p)}}\). Conclude that the mode \({{\rm{x}}^{\rm{*}}}\) is the integer satisfying \({\rm{1}} \le {{\rm{x}}^{\rm{*}}} \le {\rm{(n + 1)p}}\).

b. Show that if \({\rm{X}}\) has a Poisson distribution with parameter \({\rm{\mu }}\), the mode is the largest integer less than \({\rm{\mu }}\). If \({\rm{\mu }}\) is an integer, show that both \({\rm{\mu - 1}}\) and \({\rm{\mu }}\) are modes.

Three brothers and their wives decide to have children until each family has two female children. What is the pmf of X = the total number of male children born to the brothers? What is E(X), and how does it compare to the expected number of male children born to each brother?

A family decides to have children until it has three children of the same gender. Assuming P(B) = P(G) =\({\rm{.5}}\), what is the pmf of X = the number of children in the family?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.