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Introduction to Statistics
STATISTICS and PROBABILITY
LECTURE: PROBABILITY DISTRIBUTIONS
Prof. Dr. İrfan KAYMAZ
Atatürk University
Engineering Faculty
Department of Mechanical Engineering
P.S. These lecture notes are mainly based on
the reference given in the last page.
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objectives of this lecture Introduction to Statistics
After carefully listening of this lecture, you should be able to do the
following:
 Determine probabilities from probability mass functions and the
reverse.
 Determine probabilities from cumulative distribution functions,
and cumulative distribution functions from probability mass
functions and the reverse.
 Determine probabilities from probability density functions.
 Determine probabilities from cumulative distribution functions,
and cumulative distribution functions from probability density
functions, and the reverse.
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Random Variables
Probability
 A variable that associates a number with the outcome of a
random experiment is called a random variable.
A random variable is a function that assigns a real number to
each outcome in the sample space of a random experiment.
Particular notation is used to distinguish the random variable (rv)
from the real number. The rv is denoted by an uppercase letter,
such as X. After the experiment is conducted, the measured value
is denoted by a lowercase letter, such a x = 70. X and x are shown
in italics, e.g., P(X=x).
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Continuous & Discrete Random
Variables
Probability
 A discrete random variable is a rv with a finite (or countably
infinite) range. They are usually integer counts, e.g., number
of errors or number of bit errors per 100,000 transmitted
(rate). The ends of the range of rv values may be finite (0 ≤ x
≤ 5) or infinite (x ≥ 0).
 A continuous random variable is a rv with an interval (either
finite or infinite) of real numbers for its range. Its precision
depends on the measuring instrument.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Examples of
Discrete & Continuous RVs
Probability
 Continuous rv’s:
 Electrical current and voltage.
 Physical measurements, e.g., length, weight, time,
temperature, pressure.
 Discrete rv’s:




Number of scratches on a surface.
Proportion of defective parts among 100 tested.
Number of transmitted bits received in error.
Number of common stock shares traded per day.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Probability Distributions
Probability
 A random variable X associates the outcomes of a random
experiment to a number on the number line.
 The probability distribution of the random variable X is a
description of the probabilities with the possible numerical
values of X.
 A probability distribution of a discrete random variable can be:
 A list of the possible values along with their probabilities.
 A formula that is used to calculate the probability in response
to an input of the random variable’s value.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Example: Digital Channel
 There is a chance that a bit
transmitted through a
digital transmission
channel is received in error.
 Let X equal the number of
bits received in error of the
next 4 transmitted.
 The associated probability
distribution of X is shown
as a graph and as a table.
Probability
Figure 3-1 Probability
distribution for bits in error.
P(X =0) =
P(X =1) =
P(X =2) =
P(X =3) =
P(X =4) =
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
0.6561
0.2916
0.0486
0.0036
0.0001
1.0000
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Probability Mass Function
Probability
Suppose a loading on a long, thin beam places mass only at discrete
points. This represents a probability distribution where the
beam is the number line over the range of x and the probabilities
represent the mass. That’s why it is called a probability mass
function.
Figure 3-2 Loading at discrete points on a long, thin beam.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Probability Mass Function
Properties
Probability
For a discrete random variable X with possible values x1 ,x 2 , ... x n ,
a probability mass function is a function such that:
(1) f  xi   0
n
(2)
 f x  1
i 1
i
(3) f  xi   P  X  xi 
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Example: Wafer Contamination
 Let the random variable X denote the
number of wafers that need to be
analyzed to detect a large particle.
Assume that the probability that a wafer
contains a large particle is 0.01, and that
the wafers are independent. Determine
the probability distribution of X.
 Let p denote a wafer for which a large
particle is present & let a denote a
wafer in which it is absent.
 The sample space is:
S = {p, ap,
aap, aaap, …}
 The range of the values of X is: x = 1, 2,
3, 4, …
Probability
Probability Distribution
P(X =1) =
0.1 0.1
P(X =2) = (0.9)*0.1 0.09
P(X =3) = (0.9)2*0.1 0.081
P(X =4) = (0.9)3*0.2 0.0729
0.3439
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Cumulative Distribution Functions
 Example 3-6: From Example 3.4, we can
express the probability of three or fewer
bits being in error, denoted as P(X ≤ 3).
 The event (X ≤ 3) is the union of the
mutually exclusive events: (X=0), (X=1),
(X=2), (X=3).
 From the table:
Probability
x
0
1
2
3
4
P(X =x ) P(X ≤x )
0.6561
0.2916
0.0486
0.0036
0.0001
1.0000
0.6561
0.9477
0.9963
0.9999
1.0000
P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) = 0.9999
P(X = 3) = P(X ≤ 3) - P(X ≤ 2) = 0.0036
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Cumulative Distribution Function
Properties
Probability
The cumulative distribution function is built from the
probability mass function and vice versa.
The cumulative distribution function of a discrete random variable X ,
denoted as F ( x), is:
F  x   F  X  x    xi
xi  x
For a discrete random variable X , F  x  satisfies the following properties:
(1) F  x   P  X  x    f  xi 
xi  x
(2) 0  F  x   1
(3) If x  y, then F  x   F  y 
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Example :Cumulative Distribution
Function
Probability
 Determine the probability mass function of X from this
cumulative distribution function:
F (x) = 0.0
0.2
0.7
1.0
x < -2
-2 ≤ x < 0
0≤x <2
2≤x
PMF
f (2) = 0.2
f (0) = 0.5
f (2) = 0.3
Figure 3-3 Graph of the CDF
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Example: Sampling without
Replacement
Probability
A day’s production of 850 parts contains 50 defective parts. Two parts
are selected at random without replacement. Let the random
variable X equal the number of defective parts in the sample. Create
the CDF of X.
799
P  X  0   800

850 849  0.886
50
P  X  1  2  800

850 849  0.111
50
49
P  X  2   850
 849
 0.003
Therefore,
F  0   P  X  0   0.886
F 1  P  X  1  0.997
F  2   P  X  2   1.000
Figure 3-4 CDF. Note that F(x) is defined
for all x, - <x < , not just 0, 1 and 2.
14
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Continuous Density Functions
Probability
Density functions, in contrast to mass functions,
distribute probability continuously along an interval.
The loading on the beam between points a & b is the
integral of the function between points a & b.
Figure 4-1 Density function as a loading on a long, thin
beam. Most of the load occurs at the larger values of x.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Continuous Density Functions
Probability
A probability density function f(x) describes the
probability distribution of a continuous random
variable. It is analogous to the beam loading.
Figure 4-2 Probability is determined from the area under f(x) from a to b.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Probability Density Function
Probability
For a continuous random variable X ,
a probability density function is a function such that
(1)
f  x  0
means that the function is always non-negative.

(2)

f ( x)dx  1

b
(3)
(4)
P  a  X  b    f  x  dx  area under f  x  dx from a to b
f  x  0
a
means there is no area exactly at x.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Histograms
Probability
A histogram is graphical display of data showing a series of adjacent
rectangles. Each rectangle has a base which represents an
interval of data values. The height of the rectangle creates an
area which represents the relative frequency associated with the
values included in the base.
A continuous probability distribution f(x) is a model approximating
a histogram. A bar has the same area of the integral of those
limits.
Figure 4-3 Histogram approximates a probability density function.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Area of a Point
Probability
If X is a continuous random variable, for any x1 and x2 ,
P  x1  X  x2   P  x1  X  x2   P  x1  X  x2   P  x1  X  x2 
(4-2)
which implies that P  X  x   0.
From another perspective:
As x1 approaches x2 , the area or probability becomes smaller and smaller.
As x1 becomes x2 , the area or probability becomes zero.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Example: Hole Diameter
Probability
Let the continuous random variable X denote the diameter of a
hole drilled in a sheet metal component. The target diameter
is 12.5 mm. Random disturbances to the process result in
larger diameters. Historical data shows that the distribution
of X can be modeled by f(x)= 20e-20(x-12.5), x ≥ 12.5 mm.
If a part with a diameter larger than 12.60 mm is scrapped, what
proportion of parts is scrapped?
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Probability
Example: Hole Diameter
Figure 4-5 P  X  12.60  


20e
20 x 12.5
dx  0.135
12.6
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Cumulative Distribution
Functions
Probability
The cumulative distribution function
of a continuous random variable X is,
F  x  P  X  x 
x
 f  u  du
for    x  
(4-3)

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Example :Electric Current
Probability
For the copper wire current measurement in Exercise 4-1,
the cumulative distribution function (CDF) consists of
three expressions to cover the entire real number line.
0
x <0
F (x ) = 0.05x 0 ≤ x ≤ 20
1
20 < x
Figure 4-6 This graph shows the CDF as
a continuous function.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Example :Hole Diameter
Probability
For the drilling operation in Example 4-2, F(x) consists of
two expressions. This shows the proper notation.
F  x  0
F  x 
for x  12.5
x

20e
20 u 12.5
du
12.5
 1 e
20 x 12.5
for x  12.5
Figure 4-7 This graph shows F(x)
as a continuous function.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Density vs. Cumulative
Functions
Probability
 The probability density function (PDF) is the
derivative of the cumulative distribution
function (CDF).
 The cumulative distribution function (CDF) is
the integral of the probability density function
(PDF).
dF  x 
Given F  x  , f  x  
as long as the derivative exists.
dx
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Exercise: Reaction Time
Probability
 The time until a chemical reaction is complete (in
milliseconds, ms) is approximated by this CDF:

0
for x  0
F  x 
1  e0.01x for 0  x
 What is the PDF?
 What proportion of reactions is complete within 200
ms?
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Probability
Exercise: Reaction Time
 the PDF

dF  x  d  0
0
for x  0
f  x 
 

0.01x
0.01 x
1

e
0.01
e
for 0  x
dx
dx 
 The proportion of reactions is complete within 200
ms
P  X  200  F  200   1  e2  0.8647
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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Next Week
Probability
Most commonly used probability
functions….
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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References
Douglas C. Montgomery, George C. Runger
Applied Statistics and Probability for Engineers,
John Wiley & Sons, Inc.
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