Chapter 7: Q.7.17 (page 364)

A total of $m$ items are to be sequentially distributed among $n$ cells, with each item independently being put in a cell $j$ with probability role="math" localid="1647437056272" $p j, j = 1, . . ., n$ . Find the expected number of collisions that occur, where a collision occurs whenever an item is put into a non-empty cell.

Short Answer

Expert verified

The expected number of collisions that occur, where a collision occurs whenever an item is put into a non-empty cell,

$E [X] = m - n + \sum_{j = 1}^{n} {(1 - p_{j})}^{m}$

Step by step solution

Step 1:Concept Introduction

Given that

$X_{i} = 1$ if a collision occurs when the item is placed

$= 0$ otherwise

$X = T o t a l n u m b e r o f c o l l i s i o n s$

$= \sum_{i = 1}^{m} X_{i}$

$\Rightarrow E [X] = \sum_{i = 1}^{m} E [X_{i}]$

Step 2:Explanation

Molding on the cell in which it is set.

$E [X_{i}] = \sum_{j}^{} E [X_{i} ∣ placed in cell j] p_{j}$

$= \sum_{j} E [i causes collision ∣ Placed in cell j] p_{j}$

localid="1647435990293" $= \sum_{j}^{} [1 - {(1 - p_{j})}^{i - 1}] p_{j}$

localid="1647436010058" $= 1 - \sum_{j}^{} {(1 - p_{j})}^{i - 1} p_{j}$

Step 3:Final Answer

The close to last uniformity utilized that, restrictive on the item $i$ being set in the cell $j$ , item $i$ will cause a collision if any of the previous $(i - 1)$ items were placed in a cell $j$ , Thus,

$E [X] = m - \sum_{i = 1}^{m} \sum_{j = 1}^{n} {(1 - p_{j})}^{i - 1} p_{j}$

Exchanging the request for the summation gives

$E [X] = m - n + \sum_{j = 1}^{n} {(1 - p_{j})}^{m}$

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!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Step 1:Concept Introduction

Step 2:Explanation

Step 3:Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Probability and Statistics

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.