Friday, January 18, 2013

Calculating Sample Probabilities ( Poisson )

Ref : Design for Six Sigma Statistic (3.1.4.4)

When counting defects in a sample from a contineuous medium

The probability of observing exactly "X" defects in the sample is



Example
Let failure rate of a control systme : 30 failure per million hour

Assuming that if any one failure happen, it will be detected and repaired within 24hour.
If a second failure happpens before the 1st failure is repaired, the system will shut down unexpectedly causing great damage.



Assuming the spare part is replanished only once per year. How many spare should be keep if we want to have a 99% confidence we have enough spare to up the system in case failure



Friday, January 11, 2013

Calculating Sample Probability ( Binomail )

Ref : Design For Six Sigma Statistics (3.1.4.3)

A process engineer is performing a quality check on a large batch of incoming material. He is asuming the proable defective rate is 4% ( p=0.04) based on the past record.

Since every time he draw 5 sample ( n = 5 ) for his inspection, what would be the probability of selecting at least 1 defective part ?

Solution

If the selection has at least 1 defective part, then, it could be also containing 2, 3, 4 or 5 defective part as well.

Therefore, the probability is the sum of 5 Binomial probability expression, one for each of possibilities.




Wednesday, January 9, 2013

Bernoulli Trial

Ref : Design for Six Sigma Statistics (3.1.4.3)

A Bernoulli Trial is a simple experiement with following characteristic :
  1. Each Bernoulli trial has only two outcome :
    • Yes / No
    • Pass / Fail
  2. In an experiment with multi Bernoulli trials, the probability of each of the outcomes is the same in every trial
  3. In an experiment with multiple Bernoulli trials, all trials are mutually independent.
Suppose :
  • p = probability of selecting a defective part
  • n = part are selected from a infinite population
What is the probability of selecting exactly x defective parts ?

This situation has 'n' Bernaoulli trial with a constand probability of selecting a defect 'p'. Since the trials are indenpendent, the joint probabilities are products of individual probabilities.

Each possible sample with 'x' defective parts and (n-x) nondefective parts has a probability of

Also there arecombination of samples of size 'n'with 'x' defects.
Therefore, the probability that a 'n' sample size from a population with a constand probability fo defects 'p' with have exactly 'x' defects is


This probability model is known as Binomial Distribution