-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
Probability distributions are models used to describe the
behavior of random variables. The two types of random variables
have two types of distributions, because they are mathematically
different Discrete random variable: non-continuous distribution
Probability Mass Function Continuous random variable: continuous
distribution
Probability Density Function
Probability Distributions
2012 from "A First Course in Quality Engineering: Integrating
Statistical and Management Methods of Quality" by K.S.
Krishnamoorthi. Reproduced by permission of Taylor and Francis
Group, LLC, a division of Informa plc.
Holly Ott Quality Engineering & Management Module 2.2 11
-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
If X is a discrete random variable, a function p(x) is defined
as the probability mass function (pmf) of the random variable with
the following properties
1) p(x) 0, for all x 2) x p(x) = 1 3) p(x) = P(X = x)
Probability mass function (pmf)
2012 from "A First Course in Quality Engineering: Integrating
Statistical and Management Methods of Quality" by K.S.
Krishnamoorthi. Reproduced by permission of Taylor and Francis
Group, LLC, a division of Informa plc.
Holly Ott Quality Engineering & Management Module 2.2 12
-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
Example: a random variable X denotes the number of tails (Eagle)
when a Euro is tossed three times. Find its pmf.
Probability mass function (pmf)
Holly Ott Quality Engineering & Management Module 2.2 13
-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
Example: a random variable X denotes the number of tails when a
coin is tossed three times. Find its pmf. In a table: In a closed
form: Or, as a graph:
( ) 3,2,1,0,8
3
=
= xx
xp
0 1 2 3
p(x)
x2012 from "A First Course in Quality Engineering: Integrating
Statistical and Management Methods of Quality" by K.S.
Krishnamoorthi. Reproduced by permission of Taylor and Francis
Group, LLC, a division of Informa plc.
Probability mass function (pmf)
Holly Ott Quality Engineering & Management Module 2.2 14
-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
If X is a continuous random variable, a function f(x) is defined
with the following properties and is called the probability density
function (pdf) of X.
1) f(x) 0, for all x 2) f(x)dx = 1 3) P(a X b) = f(x)dx,
integrated over the interval from a to b
Probability density function (pdf)
2012 from "A First Course in Quality Engineering: Integrating
Statistical and Management Methods of Quality" by K.S.
Krishnamoorthi. Reproduced by permission of Taylor and Francis
Group, LLC, a division of Informa plc.
Holly Ott Quality Engineering & Management Module 2.2 15
-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
Example: a random variable is known to have the following
pdf:
( ) ( )
=
therwise o , x , x.x x, .
xf0
201020010100010
2012 from "A First Course in Quality Engineering: Integrating
Statistical and Management Methods of Quality" by K.S.
Krishnamoorthi. Reproduced by permission of Taylor and Francis
Group, LLC, a division of Informa plc.
Probability density function (pdf)
Holly Ott Quality Engineering & Management Module 2.2 16
0.10
0 5 10 20
0.100
0 5 10 200
f(x)0
-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
Example: a random variable is known to have the following
pdf:
( ) ( )
=
therwise o , x , x.x x, .
xf0
201020010100010
2012 from "A First Course in Quality Engineering: Integrating
Statistical and Management Methods of Quality" by K.S.
Krishnamoorthi. Reproduced by permission of Taylor and Francis
Group, LLC, a division of Informa plc.
Probability density function (pdf)
Holly Ott Quality Engineering & Management Module 2.2 17
0.10
0 5 10 20
0.100
0 5 10 200
f(x)0
-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
X is a discrete random variable with pmf, p(x) CDF: F(x) = P(X
x)
= p(t), for t x If X is a continuous random variable
with pdf f(x), CDF: F(x) = P(X x)
= f(t)dt, for t x
Cumulative distribution function (CDF) F(x)
1.0
0.5
x
CDF of a discrete random variable
x
F(x)
1.0
CDF of a continuous random variable
0.5
2012 from "A First Course in Quality Engineering: Integrating
Statistical and Management Methods of Quality" by K.S.
Krishnamoorthi. Reproduced by permission of Taylor and Francis
Group, LLC, a division of Informa plc.
Holly Ott Quality Engineering & Management Module 2.2 18
-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
Mean (): the weighted average, which represents the center of
gravity of a distribution (also called Expected Value)
2012 from "A First Course in Quality Engineering: Integrating
Statistical and Management Methods of Quality" by K.S.
Krishnamoorthi. Reproduced by permission of Taylor and Francis
Group, LLC, a division of Informa plc.
Mean and Variance
Holly Ott Quality Engineering & Management Module 2.2 19
If continuous with pdf = f(x):
x = x xf(x)dx
If discrete with pmf = p(x):
x = x xp(x)
-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
Variance (2): the weighted average of the squared deviations of
the values of the variable from its mean.
Standard Deviation ():
2012 from "A First Course in Quality Engineering: Integrating
Statistical and Management Methods of Quality" by K.S.
Krishnamoorthi. Reproduced by permission of Taylor and Francis
Group, LLC, a division of Informa plc.
Mean and Variance
x2 = x x( )x
2p x( ) x2 = x x( )x
2f x( )dx
! X =2!
Holly Ott Quality Engineering & Management Module 2.2 20
If continuous with pdf = f(x): If discrete with pmf = p(x):
-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
Some Important Probability Distributions
Binomial Distribution discrete Poisson Distribution discrete
Normal Distribution continuous (Other important distributions we
use in Quality Engineering are Exponential, Weibull, t-dist.,
Chi-squared dist., F-dist..) Holly Ott Quality Engineering &
Management Module 2.2 21
2012 from "A First Course in Quality Engineering: Integrating
Statistical and Management Methods of Quality" by K.S.
Krishnamoorthi. Reproduced by permission of Taylor and Francis
Group, LLC, a division of Informa plc.
-
TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
Coming Up
Lecture 2.3: Important Probability Distributions
Holly Ott Quality Engineering & Management Module 2.2 22
Foto: Thommy Weiss / pixelio.de
