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ENGR 610 Applied Statistics Fall 2007 - 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 1.25 2.54 2.55 Chapter 3 Review Measures of… Central Tendency Variation Shape Skewness Kurtosis Box-and-Whisker Plots Measures of Central Tendency Mean (arithmetic) Median Most popular (peak) value(s) - can be multi-modal Midrange Middle value - 50th percentile (2nd quartile) Mode Average value: 1 N Xi N i (Max+Min)/2 Midhinge (Q3+Q1)/2 - average of 1st and 3rd quartiles Measures of Variation Range (max-min) Inter-Quartile Range (Q3-Q1) Variance Sum of squares (SS) of the deviation from mean divided by the degrees of freedom (df) - see pp 113-5 df = N, for the whole population df = n-1, for a sample 2nd moment about the mean (dispersion) (1st moment about the mean is zero!) Standard Deviation Square root of variance (same units as variable) Sample (s2, s, n) vs Population (2, , N) Quantiles Equipartitions of ranked array of observations Percentiles - 100 Deciles - 10 Quartiles - 4 (25%, 50%, 75%) Median - 2 Pn = n(N+1)/100 -th ordered observation Dn = n(N+1)/10 Qn = n(N+1)/4 Median = (N+1)/2 = Q2 = D5 = P50 Measures of Shape Symmetry Skewness - extended tail in one direction 3rd moment about the mean Kurtosis Flatness, peakedness Leptokurtic - highly peaked, long tails Mesokurtic - “normal”, triangular, short tails Platykurtic - broad, even 4th moment about the mean Box-and-Whisker Plots Graphical representation of five-number summary Min, Max (full range) Q1, Q3 (middle 50%) Median (50th %-ile) Shows symmetry (skewness) of distribution Other Resources SPSS Tutorial at Statistical Consulting Services MathWorld http://www.stats-consult.com/tutorials.html http://mathworld.wolfram.com See Probability and Statistics Wikipedia http://en.wikipedia.org/wiki/Category:Probability_a nd_statistics Homework Problems 3.27 3.31 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. 2. 3. 4. 5. A probability P(A) for event A is between 0 (null event) and 1 (certain event) The complement of P(A) is the probability that A will not occur, and P(not-A) = 1- P(A) Two events are mutually exclusive if P(A and B) = 0 If two events are mutually exclusive, then P(A or B) = P(A) + P(B) If set of events are mutually exclusive and collectively exhaustive, then P(A ) 1 i i Rules of Probability 6. 7. 8. 9. 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. The conditional probability of B occurring, given that A has occurred, is given by P(B|A) = P(A and B)/P(A) 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) 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. E(X) X i P(X i ) i 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. X2 (X i X ) 2 P(X i ) i 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: n! P(X x | n, p) p x (1 p) nx x!(n x)! Binomial Distribution, cont’d Binomial coefficients follow Pascal’s Triangle 1 11 121 1331 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) Limit case of Binomial distribution for large n Based only on expectation value () e x P(X x | ) x! Poisson Distribution, cont’d Mean () = variance (2) = Right-skewed, but approaches symmetric bell-shape as gets large Other Discrete Probability Distributions Hypergeometric (pp 159-160) Negative Binomial (pp 162-163) 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 Number of trials (n) until xth success Binomial with last trial constrained to be a success Geometric (pp 164-165) 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) Homework Ch 4 Appendix 4.1 Problems: 4.57,60,61,64 Read Ch 5 Continuous Probability Distributions and Sampling Distributions