Probability Distributions
Module Objectives
At the end of this module, the student will be able to:
Describe the difference between continuous and discrete data
Describe and use discrete probability distributions
-Binomial Distribution
-Poisson Distribution
Describe and use continuous probability distributions
-Normal or Gaussian Distribution
The Basic Nature of Data
Two kinds of data can be used for measuring process
capability. Data which characterizes a product or process
feature in terms of it size, weight, volts. This type of data is
said to be continuous by nature. In other words, the
measurement scale can be meaningfully divided into finer and
finer increments of precision.
Another way to look at the data is to merely count the
frequency of occurrence; ., the number of times something
happens or fails to happen. Notice that such data is not
capable of being meaningfully subdivided into more precise
increments and, therefore, is said to be discrete by nature.
The Poisson and binomial models are used in connection with
this type of data.
Experiments, Sample Space & Events
In statistics, the word experiment refers to any activity that
generates a set of data. A random experiment is one that can result in different outcomes, even though it is repeated in the same manner every time.
The set of all possible outcomes of a random experiment is
called the sample space.
An event is a subset of the sample space and represents one of the outcomes.
Random Variables
A random variable is a function that assigns a real number to
each outcome in the sample space of a random experiment.
A random variable is denoted by an uppercase letter (. X),
and a corresponding lowercase letter (. x) is used to denote
a possible value of X.
A Probability Distribution of a discrete random variable is a description of the set of the possible values of the variable, along with the probability associated with each of the possible values.
The sum of all probabilities must equal to 1.
What is a Probability Distribution ?
Discrete Probability Distributions
Binomial Distribution
Poisson Distribution
Binomial Distribution
A Binomial Experiment:
An experiment consisting of n repeated trials such that
(1) the trials are independent
(2) each trial results in only 2 possible outcomes labeled
as “success” and “failure”, (designated “1” and “0”)
(3) the probability of success on every trial,
denoted as p, is the same.
$
Binomial Distribution
The distribution of the number (the random variable) resulting
in a “success” from the experiment has a Binomial Probability
Distribution as described below
P(X=x)=nCx px (1-p)n-x
(Reference)
where,
Population Mean = = np
Population Variance = 2= np (1-p)
n!=(n)(n-1)(n-2)…(3)(2)(1)
0!=1
(Ref: pp4-18 to 4-24 Basic Statistics)
Binomial Distribution
Mean 4
%
Example
The probability of a bit transmitted in error is .
Assume the transmission trials are independent.
Let X be the number of bits in error for the next 4 bits transmitted,
determine P(X=0), P(X=1), P(X=2), P(X=3) and P(X=4).
X=1 any 1 of the bits received was wrong
X=2 any 2 of the bits received was wrong
X=3 any 3 of the bits received was wrong
X=4 all 4 of the bits received was wrong
Example
Calc Probability Distributions Binomial
x P(X=x)
0
1
2
3
4
Poisson Probability Distribution
for x = 0, 1, 2 ..., & > 0
is the Poisson Probability Distribution:
Example:
(1) as the number of defects per disc.
(2) as the number of bad chips per PCBA.
(3) as the number of births per month in a city.
Mean: m = l Variance: s2 = l
(Ref: pp4-31 to 4-36 Basic Statistics)
The Poisson Distribution
%
Mean = 2
Example
A city has a history of 5 births per month, what is the
probability of having less than 3 births in the next month?
Let = 5 births per month
P (X less than 3) = P ( X = 0) + P (X = 1) + P (X = 2)
=
(Reference)
Example
Calc Probability Distributions Poisson
x P(X=x) P(Xx)
0
1
2
P(X<3) = P(X2) =
Continuous Probability Distributions
Normal Distribution
Other Distributions
Describes a wide variety of physical phenomena
Also used in many tool and capability studies
Normal Distribution or Gaussian Distribution
Population mean Population variance
When the normal distribution has = 0, 2 = 1, it is known as
a standard normal deviate, and is denoted by Z
To transform normal X distribution to Z standard normal
distribution, use
Recall how to find Normal probabilities using Minitab
Calc > Probability Distributions > Normal
Computational Equations
(Ref: pp4-37 to 4-54 Basic Statistics)
(1) Quick exercise with Standard Normal Distribution Table:
determine
(a) P(Z > )
(b) P(Z < )
(c) P(Z > ) and P(Z < )
(d) P( < Z < )
(2) Z - transformation Exercise
Consider American adult males have a population height
mean of 70 inches and a height std-deviation of 2 inches:
determine
(a) P(X<72)
(b) P(73 X 75)
(c) P(X < 66)
(d) P (X < 66) and P(X > 74)
Exercises
Normal Probability Plots
We can test whether a given data set can be described as
“normal” with a test called a Normal Probability Plot
If a distribution is close to normal, the normal probability plot
will be a straight line
Minitab makes the normal probability plot easy
» Open
» Choose: Graph > Probability Plot >
» (or) Stat > Basic Stats > Normality Test >
Produce a normal plot of each of the first 3 columns. Which
appear to be normal?
Now, graph a histogram of each
What does this reveal?
Mystery Distribution
Generate a Normal Probability Plot for the Mystery variable
in C5.
What is your conclusion? Is this a normal distribution
Mystery Distribution - Solution
Graph Histogram
Open file
Stat > Basic Statistics > Display Descriptive Statistics,
choose Graph Summary
Variable N Mean Median Tr Mean StDev SE Mean
Normal 500
Pos Skew 500
Neg Skew 500
Mystery 500
Variable Min Max Q1 Q3
Normal
Pos Skew
Neg Skew
Mystery
The t Distribution
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
df=1
df=5
df=100
Used when distribution approximates normality, but population standard deviation is not known and sample size is relatively small.
The Chi-Square Distribution
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
df=2
df=4
df=6
df=8
df=10
Common applications include
a) comparing variance of data set vs a reference value,
. is ² = 0² ?
b) comparing actual data set vs a reference distribution,
. goodness-of-fit
The F Distribution
0
1
0
1
2
3
4
5
df = 25, 120
df = 8, 12
df = 2, 12
Commonly used for comparing the variance from 2 populations
The Lognormal Distribution
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
When X’ = Ln(X), the distribution will approach normality..
Module Objectives
At the end of this module, the student will be able to:
Describe the difference between continuous and discrete data
Describe and use discrete probability distributions
-Binomial Distribution
-Poisson Distribution
Describe and use continuous probability distributions
-Normal or Gaussian Distribution
Experiment: Drawing from a deck of cards.
Random Experiment: Cannot predict type of card drawn.
Sample Space: {Clubs, Diamonds, Hearts, Spades}
Event: Card drawn is a club.
Card drawn is a diamond.
When Card is Let RV be
Club 1
Diamond 2
Heart 3
Spade 4
X is the RV to denote type of card drawn.
x is the value of the actual card drawn.
x=1 card is a club
x=2 card is a diamond
x=3 card is a heart
x=4 card is a spade
x P(X=x)
1 ¼
2 ¼
3 ¼
4 ¼
The Binomial Distribution is a very important type of discrete probability distribution. It applies to any situation in which several independent trials are conducted, each of which can have one of two possible outcomes. These two possible outcomes are usually referred to as “success” and “failure”. (Note that since there are only two possible outcomes, if the probability of success is p, then the probability of failure is 1-p)
Some examples of situations where the Binomial Distributions is applicable:
1) The number of questions a person can expect to guess right on a multiple-choice exam
2) The number of Warranty Claims that Customer Support must pay on
3) The number of parts with unacceptable appearance in a batch of aircraft skins
4) The number of defective light bulbs in a batch of bulbs
In some situations, when we have large populations and low probability of success, we can approximate the Binomial Distribution using the Poisson Distribution. In other words, the Binomial Distribution is a more general distribution with wider applicability.
Distribution - Binomial
T
Q1: How to know if a distribution is Poisson?
Arguably the most important probability distribution applicable to randomly-occurring discrete data is the Poisson Distribution. In Six Sigma practice, we can state that the Poisson Distribution can be used when the number of opportunities for non-conformance is large and the probability of an event is low. In practice, these conditions are often met.
Situations which can be modeled using the Poisson Distribution include:
1) The number of errors per page in an Aircraft Maintenance Manual
2) The number of flaws per square foot in a Kevlar roll
3) The number of calls arriving at the IT Help Desk or the Field Service office each month
Sometimes all the conditions for application of the Binomial Distribution are present, but the sample size n is too large. In such cases, the Poisson Distribution provides a good approximation of the Binomial approximation is critical to the determination of Process Yield and Process Capability and the concept of Defects per Unit, as we shall see in later section.
Distribution - Poisson
T
As we have seen previously, the Normal Distribution is a powerful tool for modeling the behavior of many phenomenon. Unfortunately, it becomes less and less accurate, or representative, of the actual sampling distribution as the size of the sample gets smaller. In other words, the Normal Distribution is best applied in the case of larger samples, of at least size 30.
The t Distribution, also known as the Student’s t Distribution, is another sampling distribution, and can be used to approximate the Standard Normal Distribution for small samples. In fact, it actually begins to closely match the Standard Normal Distribution as the sample size increases.
Like the Chi-square Distribution, the t Distribution is closely related to statistical estimation. It is also an integral part of an important statistical test known as the t test, which we will see in a later chapter.
Distribution - t
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Ch. 8 - The t Distribution
The first of the three sampling distributions we will look at is known as the Chi-Square Distribution.
Let us define a term called Chi (c) such that c=Z2 where Z is the Standard Normal Deviate. Then we may define cn=z12+Z22+…+Zn2 and we can say that cn has a Chi-Square Distribution with n degrees of freedom (df).
The degrees of freedom is the number N of independent observations in the sample (. the sample size) minus the number of statistics (mean, variance etc.) which must be estimated from the sample. Since we typically estimate only one parameter at a time, the number of degrees of freedom is usually calculated as df = N-1.
As we can see from the illustration above, the shape of the Chi-square Distribution changes radically with the number of degrees of freedom. As the number of degrees of freedom gets very large, the Chi-square Distribution begins to resemble the Normal Distribution.
Interestingly, the Mean of a Chi-square Distribution is simply n, the degrees of freedom, while the Variance is 2n.
The Chi-square distribution is closely related to statistical estimation. For example, the distribution of the sample variance of a sample drawn randomly from a Normal Distribution will follow a Chi-square distribution. It is also a key ingredient of an important statistical test known as the Chi-square test, which we will see in a later chapter.
Distribution - Chi-Square
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Ch. 25 - Chi-Square Analysis
The final sampling distribution which we look at is the F Distribution. It is used frequently in a very important analysis tool called ANOVA, or Analysis of Variance, which we will look at in great detail in the Analysis Phase of the Six Sigma Breakthrough Strategy. It is used to compare the variances of two samples from a population, to determine whether or not the samples are statistically equivalent.
To illustrate the F Distribution, consider the case of an Agent who has collected Normal process data from a machine, at two different times of the day over many weeks. For each days data, consisting of two samples (say, AM and PM), he calculates the Variance of each of the two samples, and takes the ratio of the two samples. He then calculates the next days’ Variance ratio, and so on. If he plots the distribution of those repeated Variance ratios, they will follow an F Distribution. Note that the F Distribution requires two degrees of freedom, referred to as the numerator and denominator degrees of freedom, corresponding to the size of the two samples whose Variances is being compared.
Like the Chi-square Distribution and the t Distribution, the F Distribution is closely related to statistical estimation. It is also an integral part of an important statistical test known as the F test, which we will see in a later chapter.
Distribution - F
T
Ch. 15 - Comparison of Variances
Another continuous probability distribution we will look at is the Lognormal Lognormal Distribution has a basic but critical relationship to the Normal Distribution. Simply put, a variable x has a Lognormal Distribution if logA(x) is Normally distributed. Conversely, we can say that if y=Ax , where x has a Normal distribution, then Y is said to have a Lognormal distribution.
It is very important to note that the Lognormal distribution is independent of the Base value (the A in the above equations) of the Logarithm. While the Lognormal distribution is typically applied using Base 10, it could just as easily apply in situations involving Base e (. Natural Logs) or any other Base value.
The Lognormal Distribution often is used in situations where time is the CT characteristic. Physically, it models processes where the relation of the process variable to time is progressive, or non-linear. Some examples of situations where the Lognormal distribution would be noticed:
1) The man-hours required to design minor aircraft repairs
2) If a random percent of stocked items goes bad on the shelves each time period, then the percent still good after a large number of time periods will be Lognormally distributed
3) Failure times of structural components
Data that is Lognormally distributed can usually be transformed into Normally distributed data, as we shall see in an upcoming section.
Distribution - Lognormal
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