ENGR 610
Applied Statistics
Fall 2007 - Week 3
Marshall University
CITE
Jack Smith
Overview for Today
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Review of Chapter 4
Homework problems (4.57,4.60,4.61,4.64)
Chapter 5
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Continuous probability distributions
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Uniform
Normal
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LogNormal
Exponential
Sampling of the mean, proportion
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Standard Normal Distribution (Z scores)
Approximation to Binomial, Poisson distributions
Normal probability plot
Central Limit Theorem
Homework assignment
Chapter 4 Review
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Discrete probability distributions
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Binomial
Poisson
Others
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Hypergeometric
Negative Binomial
Geometric
Cumulative probabilities
Probability Distributions
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A probability distribution for a discrete random
variable is a complete set of all possible distinct
outcomes and their probabilities of occurring, where
 P(X )  1
i
i
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
The expected value of a discrete random variable is
its weighted average over all possible values where
the weights are given by the probability distribution.
E(X)   X i P(X i )
i
Probability Distributions
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The variance of a discrete random variable is the
weighted average of the squared difference between
each possible outcome and the mean over all
possible values where the weights are given by the
probability distribution.
 X2  (X i  X ) 2 P(X i )
i
The standard deviation (X) is then the square root of
the variance.

Binomial Distribution
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Each elementary event is one of two mutually
exclusive and collectively exhaustive possible
outcomes (a Bernoulli event).
The probability of “success” (p) is constant from trial
to trial, and the probability of “failure” is 1-p.
The outcome for each trial is independent of any
other trial
n!
P(X  x | n, p) 
p x (1 p) nx
x!(n  x)!
Binomial Distribution
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Binomial coefficients follow Pascal’s Triangle
1
11
121
1331
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Distribution nearly bell-shaped for large n and p=1/2.
Skewed right (positive) for p<1/2, and
left (negative) for p>1/2
Mean () = np
Variance (2) = np(1-p)
Poisson Distribution
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Probability for a particular number of discrete events
over a continuous interval (area of opportunity)
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Assumes a Poisson process (“isolable” event)
Dimensions of interval not relevant
Independent of “population” size
Based only on expectation value ()
e  x
P(X  x | ) 
x!
Poisson Distribution, cont’d
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Mean () = variance (2) = 
Right-skewed, but approaches symmetric bell-shape
as  gets large
Other Discrete Probability Distributions
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Hypergeometric
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Negative Binomial
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Bernoulli events, but selected from finite population
without replacement
p now defined by N and A (successes in population N)
Approaches binomial for n < 5% of N
Number of trials (n) until xth success
Last selection is constrained to be a success
Geometric
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Special case of negative binomial for x = 1 (1st success)
Cumulative probabilities
P(X<x) = P(X=1) + P(X=2) +…+ P(X=x-1)
P(X>x) = P(X=x+1) + P(X=x+2) +…+ P(X=n)
Continuous Probability Distributions
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Differ from discrete distributions, in that
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Any value within range can occur
Probability of specific value is zero
Probability obtained by cumulating
bounded area under curve of Probability
Density Function, f(x)
Discrete sums become integrals
Continuous Probability Distributions
P(a  X  b) 
P(X  b) 

  E(X) 
b

b

f (x)dx
a
f (x)dx


 xf (x)dx
(Mean, expected value)


2 

2
(x


)
f (x)dx



(Variance)
Uniform Distribution
 1
b  a

f (x)  
 0




ab

2
2
(b

a)
2 
12
a x b
elsewhere
Normal Distribution
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Why is it important?
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Numerous phenomena measured on continuous
scales follow or approximate a normal distribution
Can approximate various discrete probability
distributions (e.g., binomial, Poisson)
Provides basis for SPC charts (Ch 6,7)
Provides basis for classical statistical inference
(Ch 8-11)
Normal Distribution
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Properties
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Bell-shaped and symmetrical
The mean, median, mode, midrange, and
midhinge are all identical
Determined solely by its mean () and standard
deviation ()
Its associated variable has (in theory) infinite
range (- < X < )

Normal Distribution
1
(1/ 2)[( X   x )/ x ]2
f (x) 
e
2 x
Standard Normal Distribution
1 (1/ 2)Z 2
f (x) 
e
2
where
Z
X  x
x
Is the standard normal score (“Z-score”)

With and effective mean of zero
and a standard deviation of 1
Normal Approximation to
Binomial Distribution
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For binomial distribution
x  np
 x  np(1 p)
and so
X  x
Z



x
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X  np
np(1 p)
Variance, 2, should be at least 10
Normal Approximation to
Poisson Distribution
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For Poisson distribution
x  
x  
and so
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
Z
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
X  x
x

X 

Variance, , should be at least 5
Normal Probability Plot
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Use normal probability graph paper
to plot ordered cumulative percentages,
Pi = (i - 0.5)/n * 100%, as Z-scores
- or Use Quantile-Quantile plot (see directions in
text)
- or Use software (PHStat)!
Lognormal Distribution
1
f (x) 
(X )  e



2 ln(x )
e
(1/ 2)[(ln(X )  ln(x ) )/ ln(x ) ]2
2
 ln(X )  ln(X
) /2
X  e
2
2  ln(X )  ln(X
)
2
 ln(X
)
(e
1)
Exponential Distribution
f (x)  e x
   1/ 
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Poisson, with continuous rate of change, 
Only memoryless random distribution
 X
P(x  X) 1 e
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Sampling Distribution of the Mean
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Central Limit Theorem
x  x
x  x / n
p  
p 
Continuous data
(proportion)
 (1  )
n
Attribute data
Homework
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Ch 5
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Skip Ch 6 and Ch 7
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Appendix 5.1
Problems: 5.66-69
Statistical Process Control (SPC) Charts
Read Ch 8
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Estimation Procedures