ENGR 610 Applied Statistics Fall 2007 - Week 3
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ENGR 610Applied Statistics
Fall 2007 - Week 3
Marshall University
CITE
Jack Smith
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Overview for Today Review of Chapter 4 Homework problems (4.57,4.60,4.61,4.64) Chapter 5
Continuous probability distributions Uniform Normal
Standard Normal Distribution (Z scores) Approximation to Binomial, Poisson distributions Normal probability plot
LogNormal Exponential
Sampling of the mean, proportion Central Limit Theorem
Homework assignment
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Chapter 4 Review
Discrete probability distributions Binomial Poisson Others
Hypergeometric Negative Binomial Geometric
Cumulative probabilities
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Probability Distributions A probability distribution for a discrete random
variable is a complete set of all possible distinct outcomes and their probabilities of occurring, where
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 iP(X i)i
P(X i)i
1
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Probability Distributions 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.
The standard deviation (X) is then the square root of the variance.
X2 (X i X )2P(X i)
i
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Binomial Distribution 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
P(X x | n, p) n!
x!(n x)!px (1 p)n x
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Binomial Distribution Binomial coefficients follow Pascal’s Triangle 1
1 1
1 2 1
1 3 3 1 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)
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Poisson Distribution Probability for a particular number of discrete events
over a continuous interval (area of opportunity) Assumes a Poisson process (“isolable” event) Dimensions of interval not relevant Independent of “population” size Based only on expectation value ()
P(X x | ) e x
x!
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Poisson Distribution, cont’d Mean () = variance (2) = Right-skewed, but approaches symmetric bell-shape
as gets large
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Other Discrete Probability Distributions
Hypergeometric 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
Negative Binomial Number of trials (n) until xth success Last selection is constrained to be a success
Geometric Special case of negative binomial for x = 1 (1st success)
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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)
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Continuous Probability Distributions
Differ from discrete distributions, in that 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
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Continuous Probability Distributions
P(aX b) f (x)dxa
b
P(X b) f (x)dx
b
E(X) xf (x)dx
2 (x )2 f (x)dx
(Mean, expected value)
(Variance)
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Uniform Distribution
f (x)
1
b aax b
0 elsewhere
a b
2
2 (b a)2
12
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Normal Distribution
Why is it important? 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)
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Normal Distribution
Properties 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 < )
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Normal Distribution
f (x) 1
2 x
e (1/ 2)[(X x ) / x ]2
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Standard Normal Distribution
f (x) 1
2e (1/ 2)Z 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
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Normal Approximation to Binomial Distribution
For binomial distribution
and so
Variance, 2, should be at least 10
Z X x x
X npnp(1 p)
x np
x np(1 p)
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Normal Approximation to Poisson Distribution
For Poisson distribution
and so
Variance, , should be at least 5
Z X x x
X
x
x
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Normal Probability Plot
Use normal probability graph paperto 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)!
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Lognormal Distribution
f (x) 1
2 ln(x )
e (1/ 2)[(ln(X ) ln(x ) ) / ln( x ) ]2
(X ) e ln(X ) ln(X )
2 / 2
X e2 ln(X ) ln(X )2
(e ln(X )2
1)
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Exponential Distribution
f (x) e x
1/
P(x X) 1 e X
Only memoryless random distribution
Poisson, with continuous rate of change,
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Sampling Distribution of the Mean
Central Limit Theorem
xx
x x / n
p (1 )
n
Continuous data
Attribute data
p (proportion)
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Homework Ch 5
Appendix 5.1 Problems: 5.66-69
Skip Ch 6 and Ch 7 Statistical Process Control (SPC) Charts
Read Ch 8 Estimation Procedures