/*! 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} Q117E A computer disk storage device h... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Q117E (page 139)

A computer disk storage device has ten concentric tracks, numbered \({\rm{1,2,}}...{\rm{,10}}\) from outermost to innermost, and a single access arm. Let \({{\rm{p}}_{\rm{i}}}{\rm{ = }}\)the probability that any particular request for data will take the arm to track \({\rm{i(i = 1,}}....{\rm{,10)}}\). Assume that the tracks accessed in successive seeks are independent. Let \({\rm{X = }}\)the number of tracks over which the access arm passes during two successive requests (excluding the track that the arm has just left, so possible \({\rm{X}}\) values are \({\rm{x = 0,1,}}...{\rm{,9}}\)). Compute the \({\rm{pmf}}\) of \({\rm{X}}\). (Hint: \({\rm{P}}\) (the arm is now on track \({\rm{i}}\) and \({\rm{X = j}}\))\({\rm{ = P(X = j larm nowon i)}} \cdot {{\rm{p}}_{\rm{i}}}\). After the conditional probability is written in terms of \({{\rm{p}}_{\rm{1}}}{\rm{,}}...{\rm{,}}{{\rm{p}}_{{\rm{10}}}}\), by the law of total probability, the desired probability is obtained by summing over \({\rm{i}}\).)

Short Answer

Expert verified

The \(pmf\) of \(X\) is obtained as \(\sum\limits_{{\rm{i = 1}}}^{{\rm{10}}} {{\rm{(}}{{\rm{p}}_{{\rm{i - j - 1}}}}{\rm{ + }}{{\rm{p}}_{{\rm{i + j + 1}}}}{\rm{)}} \cdot {{\rm{p}}_{\rm{i}}}} \).

Step by step solution

01

Concept Introduction

The law (or formula) of total probability is a key rule in probability theory that connects marginal probabilities to conditional probabilities. Its name comes from the fact that it expresses the entire likelihood of an outcome that can be attained through numerous different circumstances.

02

Computing the \({\rm{pmf}}\)

Everything is explained in the hint of the exercise. For\({\rm{j}} \in {\rm{\{ 0,1,}}...{\rm{,9\} }}\), the following holds 鈥�

Only the hint (conditional probability, law of total probability, summing over\({\rm{i}}\)) is used. Note that the probabilities\({{\rm{p}}_{\rm{k}}}\)are\({\rm{0}}\),\({{\rm{p}}_{\rm{k}}}{\rm{ = 0}}\), for\({\rm{k}} \notin {\rm{\{ 1,2,}}...{\rm{,10\} }}\).

The Law of Total Probability 鈥�

If \({{\rm{A}}_{\rm{1}}}{\rm{,}}{{\rm{A}}_{\rm{2}}}{\rm{,}}...{{\rm{A}}_{\rm{n}}}\) mutually exclusive, and \( \cup _{{\rm{i = 1}}}^{\rm{n}}{{\rm{A}}_{\rm{i}}}{\rm{ = S}}\), then for any event \({\rm{B}}\) the following is true are 鈥�

\(\begin{aligned}{}{\rm{P(B) = P}}\left( {{\rm{B}}\mid {{\rm{A}}_{\rm{1}}}} \right){\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ + P}}\left( {{\rm{B}}\mid {{\rm{A}}_{\rm{2}}}} \right){\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}} \right){\rm{ + \ldots + P}}\left( {{\rm{B}}\mid {{\rm{A}}_{\rm{n}}}} \right){\rm{P}}\left( {{{\rm{A}}_{\rm{n}}}} \right)\\{\rm{ = }}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{P}} \left( {{\rm{B}}\mid {{\rm{A}}_{\rm{i}}}} \right){\rm{P}}\left( {{{\rm{A}}_{\rm{i}}}} \right)\end{aligned}\)

Therefore, the expression is obtained as \(\sum\limits_{{\rm{i = 1}}}^{{\rm{10}}} {{\rm{(}}{{\rm{p}}_{{\rm{i - j - 1}}}}{\rm{ + }}{{\rm{p}}_{{\rm{i + j + 1}}}}{\rm{)}} \cdot {{\rm{p}}_{\rm{i}}}} \).

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

The number of pumps in use at both a six-pump station and a four-pump station will be determined. Give the possible values for each of the following random variables:

a. T = the total number of pumps in use.

b. X = the difference between the numbers in use atstations 1 and 2.

c. U = the maximum number of pumps in use at either Station.

d. Z = the number of stations having exactly two pumps in use.

Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is .4 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let X = the number ofpeople who arrive late for the seminar.

a. Determine the probability mass function of X. (Hint: label the three couples #1, #2, and #3 and the two individuals #4 and #5.)

b. Obtain the cumulative distribution function of X, and use it to calculate\(P\left( {2 \le X \le 6} \right)\).

There are two Certified Public Accountants in a particular office who prepare tax returns for clients. Suppose that for a particular type of complex form, the number of errors made by the first preparer has a Poisson distribution with mean value \({{\rm{\mu }}_{\rm{1}}}\), the number of errors made by the second preparer has a Poisson distribution with mean value \({{\rm{\mu }}_{\rm{2}}}\), and that each CPA prepares the same number of forms of this type. Then if a form of this type is randomly selected, the function

\({\rm{p(x;}}{{\rm{\mu }}_{\rm{1}}}{\rm{,}}{{\rm{\mu }}_{\rm{2}}}{\rm{) = }}{\rm{.5}}\frac{{{{\rm{e}}^{{\rm{ - }}{{\rm{\mu }}_{\rm{1}}}}}{\rm{\mu }}_{\rm{1}}^{\rm{x}}}}{{{\rm{x!}}}}{\rm{ + }}{\rm{.5}}\frac{{{{\rm{e}}^{{\rm{ - }}{{\rm{\mu }}_{\rm{2}}}}}{\rm{\mu }}_{\rm{2}}^{\rm{x}}}}{{{\rm{x!}}}}{\rm{ x = 0,1,2,}}...\)

gives the \({\rm{pmf}}\) of \({\rm{X = }}\)the number of errors on the selected form.

a. Verify that \({\rm{p(x;}}{{\rm{\mu }}_{\rm{1}}}{\rm{,}}{{\rm{\mu }}_{\rm{2}}}{\rm{)}}\) is in fact a legitimate \({\rm{pmf}}\) (\( \ge {\rm{0}}\) and sums to \({\rm{1}}\)).

b. What is the expected number of errors on the selected form?

c. What is the variance of the number of errors on the selected form?

d. How does the \({\rm{pmf}}\) change if the first CPA prepares \({\rm{60\% }}\) of all such forms and the second prepares \({\rm{40\% }}\)?

Refer to Chebyshev鈥檚 inequality given in Exercise\({\rm{44}}\). Calculate\({\rm{P(|X - \mu |}} \ge {\rm{k\sigma )}}\)for k\({\rm{ = 2}}\)and k\({\rm{ = 3}}\)when\({\rm{X}} \sim {\rm{Bin(20,}}{\rm{.5)}}\), and compare to the corresponding upper bound. Repeat for\({\rm{X}} \sim {\rm{Bin(20,}}{\rm{.75)}}\).

If \({\rm{X}}\) is a hypergeometric rv, show directly from the definition that \({\rm{E(X) = nM/N}}\) (consider only the case \({\rm{n < M}}\)). (Hint: Factor \({\rm{nM/N}}\) out of the sum for \({\rm{E(X)}}\), and show that the terms inside the sum are of the form \({\rm{h(f;n - 1,M - 1,N - 1)}}\), where \({\rm{y = x - 1}}\).)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

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.