ENGR 610 Applied Statistics Fall 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.

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.

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

Binomial Distribution Binomial coefficients follow Pascal’s Triangle Distribution nearly bell-shaped for large n and p=1/2. Skewed right (positive) 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 ( )

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 x th success Last selection is constrained to be a success Geometric Special case of negative binomial for x = 1 (1 st 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 (Mean, expected value) (Variance)

Uniform Distribution

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

Standard Normal Distribution where 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

Normal Approximation to Poisson Distribution For Poisson distribution and so Variance,, should be at least 5

Normal Probability Plot Use normal probability graph paper to plot ordered cumulative percentages, P i = (i - 0.5)/n * 100%, as Z-scores - or - Use Quantile-Quantile plot (see directions in text) - or - Use software (PHStat)!

Lognormal Distribution

Exponential Distribution Only memoryless random distribution Poisson, with continuous rate of change,

Sampling Distribution of the Mean Central Limit Theorem Continuous data Attribute data (proportion)

Homework Ch 5 Appendix 5.1 Problems: Skip Ch 6 and Ch 7 Statistical Process Control (SPC) Charts Read Ch 8 Estimation Procedures