ENGR 610Applied Statistics

Fall 2007 - Week 3

Marshall University

CITE

Jack Smith

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

Chapter 4 Review

Discrete probability distributions Binomial Poisson Others

Hypergeometric Negative Binomial Geometric

Cumulative probabilities

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

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

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

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)

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!

Poisson Distribution, cont’d Mean () = variance (2) = Right-skewed, but approaches symmetric bell-shape

as gets large

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)

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

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

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)

Uniform Distribution

f (x)

1

b aax b

0 elsewhere

a b

2

2 (b a)2

12

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)

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 < )

Normal Distribution

f (x) 1

2 x

e (1/ 2)[(X x ) / x ]2

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

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)

Normal Approximation to Poisson Distribution

For Poisson distribution

and so

Variance, , should be at least 5

Z X x x

X

x

x

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)!

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)

Exponential Distribution

f (x) e x

1/

P(x X) 1 e X

Only memoryless random distribution

Poisson, with continuous rate of change,

Sampling Distribution of the Mean

Central Limit Theorem

xx

x x / n

p (1 )

n

Continuous data

Attribute data

p (proportion)

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