ENGR 610 Applied Statistics Fall Week 2 Marshall University CITE Jack Smith

Overview for Today Homework problems 1.25, 2.54, 2.55 Review of Ch 3 Homework problems 3.27, 3.31 Probability and Discrete Probability Distributions (Ch 4) Homework assignment

Homework problems

Chapter 3 Review Measures of… Central Tendency Variation Shape Skewness Kurtosis Box-and-Whisker Plots

Measures of Central Tendency Mean (arithmetic) Average value: Median Middle value - 50 th percentile (2 nd quartile) Mode Most popular (peak) value(s) - can be multi-modal Midrange (Max+Min)/2 Midhinge (Q 3 +Q 1 )/2 - average of 1 st and 3 rd quartiles

Measures of Variation Range (max-min) Inter-Quartile Range (Q 3 -Q 1 ) Variance Sum of squares (SS) of the deviation from mean divided by the degrees of freedom (df) - see pp df = N, for the whole population df = n-1, for a sample 2 nd moment about the mean (dispersion) (1 st moment about the mean is zero!) Standard Deviation Square root of variance (same units as variable) Sample (s 2, s, n) vs Population (  2, , N)

Quantiles Equipartitions of ranked array of observations Percentiles Deciles - 10 Quartiles - 4 (25%, 50%, 75%) Median - 2 P n = n(N+1)/100 -th ordered observation D n = n(N+1)/10 Q n = n(N+1)/4 Median = (N+1)/2 = Q 2 = D 5 = P 50

Measures of Shape Symmetry Skewness - extended tail in one direction 3 rd moment about the mean Kurtosis Flatness, peakedness Leptokurtic - highly peaked, long tails Mesokurtic - “normal”, triangular, short tails Platykurtic - broad, even 4 th moment about the mean

Box-and-Whisker Plots Graphical representation of five-number summary Min, Max (full range) Q 1, Q 3 (middle 50%) Median (50 th %-ile) Shows symmetry (skewness) of distribution

Other Resources SPSS Tutorial at Statistical Consulting Services MathWorld See Probability and Statistics Wikipedia nd_statistics

Homework Problems

Chapter 4 Probability Introduction to Probability Rules of Probability Discrete Probability Distributions Probability Distributions Binomial Distribution Poisson Distribution Hypergeometric, Negative Binomial, Geometric Distributions

Introduction to Probability Probability - numeric value representing the chance, likelihood, or possibility that an event will occur Classical, theoretical Empirical Subjective Elementary event - a distinct individual outcome Event - a set of elementary events Joint event - defined by two or more characteristics

Rules of Probability 1. A probability P(A) for event A is between 0 (null event) and 1 (certain event) 2. The complement of P(A) is the probability that A will not occur, and P(not-A) = 1- P(A) 3. Two events are mutually exclusive if P(A and B) = 0 4. If two events are mutually exclusive, then P(A or B) = P(A) + P(B) 5. If set of events are mutually exclusive and collectively exhaustive, then

Rules of Probability 6. If two events are not mutually exclusive, then P(A or B) = P(A) + P(B) - P(A and B), where P(A and B) is the joint probability of A and B. 7. The conditional probability of B occurring, given that A has occurred, is given by P(B|A) = P(A and B)/P(A) 8. If two events are independent, then P(A and B) = P(A) x P(B) and P(A) = P(A|B) and P(B) = P(B|A) 9. If two events are not independent, then P(A and B) = P(A) x P(B|A)

Probability Distributions A probability distribution for a discrete random variable is complete set of all possible distinct outcomes and their probabilities of occurring, whose sum is 1. 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 (frequencies) are given by the probability distribution. The standard deviation (  X ) is then the square root of the variance.

Binomial Distribution Each elementary event is a Bernoulli event, with one of two mutually exclusive and collectively exhaustive possible outcomes. 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 The proportion of trials resulting in x successes, out of n trials, with a constant probability of p, is given by:

Binomial Distribution, cont’d 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) Limit case of Binomial distribution for large n 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 (pp ) Bernoulli events, but selected from finite population without replacement p  A/N, where A number of successes in population N Approaches binomial for n < 5% of N Negative Binomial (pp ) Number of trials (n) until x th success Binomial with last trial constrained to be a success Geometric (pp ) 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)

Homework Ch 4 Appendix 4.1 Problems: 4.57,60,61,64 Read Ch 5 Continuous Probability Distributions and Sampling Distributions