Quality Sampling and Reliability

New Uses for the Poisson Distribution

John J Heldt

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9781574442410
• eBook ISBN: 9781000162226
• Topic: Business
• Subtopic: Business General
• Index: Business

As a mathematical model for determining the probable number of outcomes, the new Poisson Distribution tables have long been an easier tool to use for reliability analyses. Longtime quality professional, inventor, and consultant John J. Heldt now makes the Poisson Table even more useful-creating two new tables (available only in this book) with the Poisson terms rearranged for further ease of estimation.

Quality Sampling and Reliability: New Uses for the Poisson Distribution simplifies the steps involved with reliability testing; Mean Time Between Failure (MTBF) assessment; advantages and risks involved in reliability life testing; and an example of methodology for tracking the MTBF for products in the field.

In addition to the tried-and-true Standard Poisson table, used to review conventional Poisson uses, Heldt's two variations yield these results:

Estimations of product Mean Time Between Failures (MTBFs), based on life tests-including the 90%, 80% or 60% envelop for any MTBFs that have been derived

Development of the Operating Characteristic Curves for Life testing-showing the risks and advantages of any test used to assure the product MTBF is not varying in a detrimental manner

Written for easy comprehension, with numerous illustrations, Quality Sampling and Reliability: New Uses for the Poisson Distribution will help quality professionals, engineers, instructors and students alike in their reliability testing tasks.

1

APPLICATIONS USING THE CONVENTIONAL POISSON TABLE

1.1 THE CUMULATIVE POISSON DISTRIBUTION TABLE⁴

Before we can develop applications using the Poisson table, we need to show what PT-1 (see Appendix 1) can do. Those already familiar with the use of this table can skip through this section.

The first page of the cumulative Poisson distribution is shown in Figure 1. The numbers across the top of the table designate the number of events, identified as c numbers. For sample plan purposes, these c numbers can be considered the number of failures.

The expected values⁵ (np′ are in the column at the left edge of the chart. The n part of this np′ represents the “sample size.” The p′ represents the inherent fraction defective in the lot under test.

Probabilities that c or fewer events will occur are shown in the body of this table. For sample plans, these probabilities are the probability of acceptance (Pa).

Figure 1 Cumulative Poisson Distribution (the probability of c or fewer occurrence of an event that has an expected value equal to np′).

1.2 HOW TO FIND INDIVIDUAL POISSON PROBABILITIES FROM THE CUMULATIVE POISSON TABLE

Figure 2A shows how individual probabilities can be found, using this Poisson table. In this example, the fraction (0.916) is found where the “c = 5” column⁶ intersects the “np′ = 3.00” line.⁶ in other words,

Figure 2A Cumulative Poisson Distribution (the probability of c or fewer occurrence of an event that has an expected value equal to np′).

$$0.916=\text{P}\left(5\right)+\text{P}\left(4\right)+\text{P}\left(3\right)+\text{P}\left(2\right)+\text{P}\left(1\right)+\text{P}\left(0\right)$$

It also means that 0.916 is the probability of finding 5 or fewer bad parts (failures) in an n = 100 sample, when the true fraction (p′)⁷ of defectives in the lot is 0.03 (i.e., np′ = 100 × 0.03 = 3.0).

This example shows the meaning of cumulative in the name of this table. Cumulative means that all of the Pa values listed in the column under the c = 5 column head are the addition of all of the probabilities up to and including the probability of finding 5 failures. Thus, each Pa shown in the c = 0 column is the probability of finding exactly 0 failures, since there are no previous probabilities to add. All of the Pa values in the c = 1 column are the probabilities of finding 1 or 0 failures. All Pa values in the c = 2 column are the probabilities of finding 2, or 1, or 0 failures in the sample, and so on.

Figure 2B How to find the probability of exactly five events when the expected value (np′) equals 3.00.

Figure 2B shows how subtraction is used to find the Pa for an exact number of failures from this table. To illustrate, the Pa fraction (0.815) found where the c = 4 column intersects the np′ = 3.00 line is the probability of finding 4 or 3, or 2, or 1, or 0 failures when the sample size (n) multiplied by the true fraction of defects (p′) in the lot (np′) is equal to 3.00. To find the probability of finding exactly 5 failures, we subtract P(4 or less) from P(...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• The Author
• 1 Applications Using the Conventional Poisson Table
• 2 Applications Using the Probability of Acceptance Poisson Table
• 3 Quick and Easy Poisson Applications
• 4 Double Sampling Using Poisson Applications
• 5 Applications Using the Poisson Confidence Level Table
• 6 Applications Using the Poisson Probability of Acceptance Table
• 7 MTBF Tracking Program Based on Field Data Using the Poisson Confidence Level Table
• Appendix 1: Cumulative Poisson Distribution: Poisson PT-1
• Appendix 2: Probability of Acceptance: Poisson PT-2
• Appendix 3: Confidence Level: Poisson PT-3
• Index