Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Q. 7.8 (page 364)

An arriving plane carries r families. A total of njof these families have checked in a total of jpieces of luggage, ∑inj=r. Suppose that when the plane lands, the N=∑ijnjpieces of luggage come out of the plane in a random order. As soon as a family collects all of its luggage, it immediately departs the airport. If the Sanchez family checked in j pieces of luggage, find the expected number of families that depart after they do.

Short Answer

Expert verified

The expected value is equal to∑ini·ii+j.

Step by step solution

01

Given Information

An arriving plane carries r families as∑inj=r.

02

Explanation

The number of families that depart after Sanchez family can be described as X=∑k=1r-1Ik

Observe that kth family leaves later if and only if all Sanchez's luggages have been placed before the last luggage of kth family. So, we can only consider i+jluggages and the probability that the last luggage comes from the set that has imembers is equal to

PIk=1=ii+j

03

Explanation

Now, we group random variables Ikaccording to the number of luggages they posses. We have

X=∑i∑ki=1niIki

Therefore, E(X)=∑i∑ki=1niEIki=∑i∑ki=1niPIki=1

=∑ini·ii+j.

04

Final answer

The expected value is equal to∑ini·ii+j.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

An urn has m black balls. At each stage, a black ball is removed and a new ball that is black with probability p and white with probability 1−p is put in its place. Find the expected number of stages needed until there are no more black balls in the urn. note: The preceding has possible applications to understanding the AIDS disease. Part of the body’s immune system consists of a certain class of cells known as T-cells. There are2 types of T-cells, called CD4 and CD8. Now, while the total number of T-cells in AIDS sufferers is (at least in the early stages of the disease) the same as that in healthy individuals, it has recently been discovered that the mix of CD4 and CD8 T-cells is different. Roughly 60 percent of the T-cells of a healthy person are of the CD4 type, whereas the percentage of the T-cells that are of CD4 type appears to decrease continually in AIDS sufferers. A recent model proposes that the HIV virus (the virus that causes AIDS) attacks CD4 cells and that the body’s mechanism for replacing killed T-cells does not differentiate between whether the killed T-cell was CD4 or CD8. Instead, it just produces a new T-cell that is CD4 with probability .6 and CD8 with probability .4. However, although this would seem to be a very efficient way of replacing killed T-cells when each one killed is equally likely to be any of the body’s T-cells (and thus has probability .6 of being CD4), it has dangerous consequences when facing a virus that targets only the CD4 T-cells

Let X1,X2,…be a sequence of independent random variables having the probability mass functionPXn=0=PXn=2=1/2,n≥1

The random variable X=∑n=1∞Xn/3nis said to have the Cantor distribution.

Find E[X]andVar(X).

Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards that need to be turned face up in order to obtain

(a) 2 aces;

(b) 5 spades;

(c) all 13 hearts.

For a group of 100 people, compute

(a) the expected number of days of the year that are birthdays of exactly 3 people;

(b) the expected number of distinct birthdays.

Show that Y=a+bX, then

ρ(X,Y)=+1    ifb>0−1    ifb<0
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.

Sign-up for free