6西格玛应用

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Probability Distributions
What is a Probability Distribution?
Event
Example 1: Some events from tossing of a dice.
In many cases, it is useful and convenient to define the aspect of the experiment we are interested in by denoting the event of interest with a symbol (usually an uppercase letter), e.g.:
many shafts from the same process, the outcomes will generally be different, and cannot be predicted in advance with total certainty.
An experiment which can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment.
Z Distribution t Distribution c2 Distribution F Distribution
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Probability Distributions
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Probability Distributions
What is a Probability Distribution?
Random Variable
From a same experiment, different events can be derived depending on which aspects of the experiment we consider important.
Measuring shaft Pins
X = Parts out of specs.
(LSL = 8 mm, USL = 10 mm)
LSL
0..,7.99998, 7.99999, 8, 8,00001,…, 9.99999, 10, 10.00001, 10.00002, …
What will be taught….
Tools to be taught
Parameter Estimation Hypothesis Testing ANOVA
Inferential Statistics
comprises those methods concerned with the analysis of a subset of data leading to predictions or inferences about the entire set of data
Experiment, Sample Space, Event Random Variable
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Probability Distributions
What is a Probability Distribution?
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Probability Distributions
What is a Probability Distribution?
Sample Space
The collection of all possible outcomes of an experiment is called its sample space.
8
Experiment generates continuous data
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Probability Distributions
What is a Probability Distribution?
Random Experiment If we throw the dice again and again, or produce
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Probability Distributions
We are crossing the bridge now ..
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data
Inspecting
(b) for stain
Accept Experiment generates
marks Pins
Stains
Reject
attribute data
(c) Measuring shaft Pins
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10.53 mm 10.49 mm 10.22 mm 10.29 mm 11.20 mm ……
What do we mean by inference of data?
To appreciate the notion of a probability distribution, we need to review various fundamental concepts related to it:
Charts
Loct Histogram Bar Chart Trend Chart Tables
Frequency Distribution
Mean Median Mode Dispersion
Range Standard Deviation Variance Shape
{1, 2, .., 6}
- Tossing of a dice
{Pass, Fail}
- Inspecting for stain marks
{All possible values
- Measuring shaft
between 0 and 10mm)
Event
An outcome, or a set of outcomes, from a random experiment is called an event, i.e. it is a subset of the sample space.
Descriptive Statistics
comprises those methods concerned with collecting and describing a set of data so as to yield meaning information
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Let X be the event “the number of a dice is odd”. Let W be the event “the shaft is within specs.”.
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Probability Distributions
What is a Probability Distribution?
Random Variable
We have defined a function that assigns a real number to an experimental outcome within the sample space of the random experiment.
Probability Distributions
Statistics — Overview
Statistics
Descriptive Statistics
Inferential Statistics
Charts & Tables
Numerical Measures
Parameter Estimation Hypothesis Testing
Probability Distributions
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Probability Distributions
What we have learned thus far ..
Tools Learned
Descriptive Statistics Capability Study Process Mapping Pareto Chart Fishbone Diagram FMEA
Event 1: the outcome is a diameter > mean
E1= {x > m}
Event 2: the outcome is a part failing specs.
E2 = {x < LSL, x > USL}
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Probability Distributions
Learning Objectives
What is a Probability Distribution?
Experiment, Sample Space, Event Random Variable, Probability Functions (pmf, pdf, cdf)
Experiment
An experiment is any activity that generates a set of data, which may be numerical or not numerical.
(a) Throwing a dice
Experiment generates {1, 2, .., 6} numerical / discrete
Discrete Distributions
Binomial Distribution Poisson Distribution
Continuous Distributions
Normal Distribution Exponential Distribution
Sampling Distributions
Point Estimate Interval Estimate
Parametric Methods Nonparametric Methods
Skewness Kurtosis
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Probability Distributions
What is a Probability Distribution?
As we progress from description of data towards inference of data, an important concept is the idea of a probability distribution.
Event 1: the outcome is an odd number
E1 = {1, 3, 5}
Event 2: the outcome is a number > 4
E2 = {5, 6}
Example 2: Some events from measuring shaft :