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Documents Flashcards Grammar checker. Math Statistics And Probability A. Unless otherwise specified, use Inspection Standard II. Number of pieces accepted.

Number of pieces rejected. Use first sampling plan below arrow. Use first sampling plan above arrow. The lot is acccepted when failures are grater than Ac but less then Re. Subsequent lots are, however, subject to normal inspection. The number of samples is determined according to the limits of the binomial distribution indexes.

Values in parentheses indicate the minimum quality required to have 19 out of 20 lots pass on average. This approximates AQL values. Will not be passed. Single sampling 2 1. If two-sided probability of distribution is considered, then the above values, respectively, correspond to 2.

This is referred to as upper 20 percent point.

### A.1 AQL Sampling Table

Displacement mm [single amplitude] v: The Japan society of machinery water vapor tables new edition Note: The probability that a device will fail before the time ti is known as the failure or non-reliability distribution function f ti. Also, the probability that a device will not fail before time ti is the reliability function R ti.

These functions are shown in figure B. When equipment can be repaired by renewing a failed device, the mean value of the interval that operation is possible between occurrences of failures is know as the MTBF Mean Time Between Failures. If the operating time between subsequent failures of a device throughout its life until discarded is given by t1, t2, MTBF known after the life of equipment is of no practical use.

Therefore, the MTBF for a truncated portion of the life of the equipment up to the time T0 is estimated using the following expression: In general, once a semiconductor device has failed, it cannot be repaired and used again. That is, it is a non-maintainable component.

This is analogous to the fact that the remaining life of an adult is not necessarily equal to the expected lifetime of a new born child minus the adult’s actual age.

For example, if we say that a given semiconductor device has a failure rate of 10 FIT, this means that his device fails for every component hours. This, however, is not equivalent to saying the device lifetime is hours. This is because the denominator of the defining expression equation IX component hours jus not refer to any particular device.

Such a system is known as a series system of redundancy of 0 figure B. This is a parallel system with a redundancy of n -1 figure B. In this case as well, the failures of individual devices are taken to be mutually independent.

This model consists of m units connected in series, which i-th unit has ni devices connected in parallel. Moreover, we consider the system in 0620 B. For the system in figure B. The reliability function R t for this system is given by the Poisson partial sum.

Let jos assume that the device will fail when the characteristic value changes to XL. If the amount of change in the characteristic value is found to be accelerated by thermal stress, in many cases the Arrhenius chemical reaction kinetics model can be applied to this phenomenon.

In chemical reactions, if molecules reach the temperature above which they may react the activation hisa reaction occurs. The higher the temperature of the molecules, the higher becomes their energies, and so increasing the temperature quickens reactions.

Arrhenius expressed the chemical reaction rate, K, experimentally as follows: The reaction rate K using this model is given by the following expression: As the device is subjected to more and more cycles of intermittent operation nas shown in figure B. The level of degradation of the 060 can be expressed as a function of the leakage current i.

If we take this current i as the device characteristic X discussed in B. We can, however, make a generalization about the state function f X and the characteristic value X that defines the state. Jie is based on equation IX The relationship between activation energy and the acceleration factor is shown in figure B.

The possible test results are limited to either 1 “failure or defect”or 2 “no failure or acceptance,” with no possibility of such results as “pending decision” or “exception acceptance” allowed. The values of p and q will be uniform for all test results.

Each test result is independent from one another. This discrete model is termed the Bernoulli trial or sampling. For ease of understanding we have chosen “failure” and “no failure” for results 1 and 2respectively. The fundamental condition of the Bernoulli trial is that results are only two types and they are definitely identified.

Such a phenomenon occurs with some kis. This probability is described by the binomial probability distribution fBin x, n, p. Such a probability is a typical example of the binomial probability distribution.

After n times of tests, any one of the results is E1, E2, The result Ei occurs with the probability pi.

## JIS Z 0602:1988

Hence, results fall into m classes E1, E2, This is known as the multinomial distribution. By inspecting n randomly sampled devices, x defective devices are detected with the probability fH – geo N, R, n, x. Different ways are available for sampling n devices from N devices.

If the sampled device is returned each time to choose a device from N constantly, or using replacement operations in other words, we will obtain a binomial distribution. If the sampled device is not 06002 non-replacement0062 will have a hypergeometric distribution.

If the original population is large, however, the fraction defective obtained through a sampling inspection can be approximated by the binomial distribution. If the number of defective devices x detected in a sample of n devices does not exceed c, the entire lot is considered to have passed R inspection.

How the lot acceptance rate changes is illustrated by the operation characteristic curve OC curve in figure B. R Lot acceptance rate 1. The risk of this rejection implies the producer’s loss, so the risk is referred to as producer’s risk. This is known as consumer’s risk. The AQL plan measures the lot whose fraction defective is p1 as having the lowest acceptable quality level.

By considering the probability of detecting the first failure at time t, the relation of exponential and geometric distributions can be identified. Let us assume that floating dust causes an average of r mask defects on the surfaces of silicon wafers of a given lot.

The area of the wafer surface S is divided into many portions. From one end of the wafer, each portion is inspected with a microscope. Since the location of the dust particles on the wafer surface is js, we can assume that mask defects occur completely at r random.

A mask defect can be detected at the same level of expectation at the inspection area t and at the number of inspection cycles x figure B. The average probability of finding a mask defect within the area C approaches the probability that a mask defect exists on a point on the surface of the wafer. The probability of defective items produced in the manufacturing process is p. We can find a relation between the Pascal distribution and the binomial distribution if we interpret the Pascal distribution as a Bernoulli trial comprising n – 1 tests with phenomenon 1 invariably occurring at the last n-th time after x – 1 times of phenomenon 1.

Noting the relation expressed by Eqs. IX through IX, the Pascal and Gamma distributions are the discrete and continuous probability distributions for the same probability model. This is consistent with the fact that the geometric and exponential distributions are discrete and continuous distributions, respectively IX The Gamma distribution is a failure probability density function if failure occurrence is considered to follow the Poisson process.

We deal with this issue nis B. We cannot expect that there will be no collision for some time since one has just occurred. Similarly, we cannot say that a cosmic 0062 will soon collide with a semiconductor device because there have been no collisions for a moment. This is an example of cases in which we can 0062 that a rare phenomenon will occur with some expectation if the period of observation is sufficiently long or the population to be observed is sufficiently large.

Here we assume that the phenomenon does not occur twice or more at the same instant, and further that the probability of the phenomenon occurring is constant. Such a probability process is known as the Poisson process. Detailed discussions of the Poisson distribution are given below. This probability is expressed as follows using equation IX for the Poisson distribution. In this case, the failure probability density function becomes equation IX for the Gamma distribution.

## A.1 AQL Sampling Table

This is explained as follows. Using a fixed value, in equation IX, for the number of damages k received before failure, consider the failure distribution function F t1, k as a function of time t1. The Poisson distribution approximates the binomial probability distribution if the population is large and the phenomenon occurs with a low probability.