Poisson Distribution - Basic Idea

One thing I remember from the "Lot Acceptance Plans" class is the concept of defectives and defects.  The instructor said that the two are different.

1. Defectives mean the products fail to meet the intended propose.
2. Defects mean some units of the product fail the specific requirements.

Thus, a defective contain one or more defects. Also, not all defectives cause defects.

Statistically, the distribution of “defectives” is binomial but of “defects” is Poisson.

I did not quite get it right away. That was all the instructor said in the class. Later on, I dug up a bit and get something more as follows.

Example1. Given the percent of defective is p (may be from historical records), what is the probability that a lot of n products contains d defectives?  The answer is $P(d)=\frac{n!}{(n-d)!d!}{{p}^{d}}{{(1-p)}^{n-d}}$ , where d is a binomial random variable.

Example2. Given the mean of defects in a sample of 1,000 light bulbs is 5, what is the probability that we can find 10 defects from the lot of the same size? The answer is $P(10) = \frac{{{5^{10}}}}{{10!}}{e^{ - 5}}=0.181$

When n is large and p is small, then $\lambda = np$ is moderate (around 10?), Poisson distribution $P(i) = \frac{{{\lambda ^i}}}{{i!}}{e^{ - \lambda }}$ can be used to approximate the binomial distribution.

For the rule of thumb:

The sample size n should be equal to or larger than 20 and the probability of a single success, p, should be smaller than or equal to .05. If n >= 100, the approximation is excellent if np is also <= 10.

ref: Count Control Charts