AGENDA Beta distribution Weibull distribution Homework Review BETA DISTRIBUTION Continuous random variable that is between 0 and 1 Frequently used in engineering project management Probability that some task will be completed on time Parkinson’s law Student syndrome Parameters Alpha Beta BETA DISTRIBUTION BETA DISTRIBUTION WEIBULL DISTRIBUTION Found when a random variable is normally distributed, but mean – some number of standard deviations cannot be less than 0 Parameters Alpha Beta WEIBULL DISTRIBUTION WEIBULL DISTRIBUTION HOMEWORK 5.20, 5.24, 5.26 REVIEW Descriptive methods Basic summary statistics Probability concepts Permutations and combinations Discrete distributions Continuous distributions Test Format DESCRIPTIVE METHODS Pareto diagrams Dot diagrams Graphs of frequency distributions Ogive Stem and leaf diagrams PARETO DIAGRAMS Special bar chart Based on the Pareto 80-20 Principle Ordered in descending order of interest Allows attention to be directed on most important areas Frequently include cost related data DOT DIAGRAMS Visually summarizes individual data Check for unusual patterns Easily identifies outliers Differences in data sources Machines Personnel Materials GRAPHS OF FREQUENCY DISTRIBUTIONS Histogram of cell observations Horizontal or vertical Size is based on observations in each cell OGIVE Graph of cumulative distribution STEM AND LEAF DISPLAYS Smaller sets of data Does not lose any information Class, as well as, actually data values Data values are listed to the right of the classes BASIC SUMMARY STATISTICS Mean Variance Standard Deviation Median Quartiles Mode Min Max PROBABILITY CONCEPTS Sample spaces and Venn Diagrams Permutations and combinations… Probability… Conditional probability… Mathematical expectation… VENN DIAGRAMS PERMUTATIONS AND COMBINATIONS Permutations When solution element order matters Combinations When solution element order does not matter PROBABILITY n equally likely possibilities s successes Probability of success is s/n Probability examples Getting a 6 on a dice roll = 1/6 Drawing an ace in a deck of cards = 4/52 CONDITIONAL PROBABILITY What is the probability of event A given event B has occurred Example Probability of high fidelity =0.81 Probability of both high fidelity and high selectivity=0.18 Probability of high selectivity with high fidelity=0.18 / 0.81 MATHEMATICAL EXPECTATION Mathematical Expectation For amounts a1, a2, …ak With probabilities p1, p2, …pk E=a1*p1+a2*p2+…ak*pk Example 1000 raffle tickets Winning prize is $500 E for a single ticket=1/1000 * $500=$0.50 DISCRETE DISTRIBUTIONS Binomial Distribution Hypergeometric Distribution Poisson Process and Distribution Geometric Distribution Multinomial Distribution BINOMIAL DISTRIBUTION Consists of a series of Bernouilli trials Bernoulli trials have only two possible outcomes for each trial Success = p Failure = 1-p The probability for success or failure is the same percentage for each Bernouilli trial for all the trials Outcomes of different Bernouilli trials are independent Prior outcomes have no impact on future outcomes Fixed number of Bernouilli trials, n HYPERGEOMETRIC DISTRIBUTION Situations involving sampling without replacement Lot sizes N with fraction successes (defective), a Choose a sample of size n Probability of obtaining x successes in sample POISSON PROCESS AND DISTRIBUTION Poisson process Physical process wholly or in part controlled by some sort of chance mechanism Occurrences do not come at regular intervals The arrival of customers into a system The breakdown of manufacturing machinery Poisson distribution Distribution of these “random” events POISSON APPROXIMATION TO BINOMIAL PROBABILITIES Some situations may make calculating binomial probabilities cumbersome Can approximate binomial probabilities with a Poisson approximation Probabilities successes (defectives) is “small” Lots are “large” Lambda = n * p GEOMETRIC DISTRIBUTION Probability of getting the first success after a number of failures x is the attempt number of the first success p is the probability of a given success MULTINOMIAL DISTRIBUTION Extension of the binomial distribution Can now have more than two mutually exclusive outcomes Parameters n number of independent trials xi number of a specific outcome pi probability of a specific outcome xi CONTINUOUS DISTRIBUTIONS Normal Distribution Uniform Distribution Lognormal Distribution Gamma distribution Exponential distribution Beta distribution Weibull distribution Simulation application NORMAL DISTRIBUTION Most important distribution Frequently encountered in both Nature Man made processes AKA Bell curve Gaussian distribution Particular properties Symetrical about the mean 68% of observations are found +- 1 std from mean 95% of observations are found +- 2 std from mean 99.7% of observations are found +-3 std from mean UNIFORM DISTRIBUTION All observations between the minimum and maximum specified values are equally likely Parameters Alpha=minimum value Beta=maximum value LOGNORMAL DISTRIBUTION Random variable whose logarithm is normally distributed Parameters Alpha Beta These don’t have meaning in the same sense as the uniform distribution Their values determine the shape of the distribution GAMMA DISTRIBUTION Used in reliability calculations Parameters Alpha Beta If alpha=1, gamma distribution degenerates to the exponential distribution EXPONENTIAL DISTRIBUTION Gamma distribution with alpha parameter=1 Related to random poisson processes Poisson distribution was number of observations in a given period of time Exponential distribution is the time between observations for a poisson process BETA DISTRIBUTION Continuous random variable that is between 0 and 1 Frequently used in engineering project management Probability that some task will be completed on time Parkinson’s law Student syndrome Parameters Alpha Beta WEIBULL DISTRIBUTION Found when a random variable is normally distributed, but mean – some number of standard deviations cannot be less than 0 Parameters Alpha Beta TEST FORMAT Completely multiple choice 4 choices One is none of the above Guess has 25% probability of correct answer 0 partial credit Questions are independent Errors in one do not carry over into subsequent problems 1st section – 2 points each General knowledge of probability and statistics Does not require indepth calculations 2nd section – 4 points each Requires calculations Many responses are based on wrong approaches Devise a test strategy