AGENDA Probability Random Variables Binomial Distribution Hypergeometric Distribution Mean and variance of a probability distribution Poisson Process and Distribution Geometric Distribution Multinomial Distribution Simulation RANDOM VARIABLES Random variables A function that assigns a value to a possible outcome Discrete – This chapter Continuous – Next chapter Probability distribution… List of possible values and probability Cumulative Distribution Function… Probability a random variable is less than or = x PROBABILITY DISTRIBUTION List of possible values and probability f(x)=… f(x)s sum to 1.0 f(1)=0.26, F(2)=0.50, F(3)=0.22, F(4)=0.02 CUMULATIVE DISTRIBUTION FUNCTION AKA Distribution Function Probability a random variable is less than or = x F(x)=… F(1)=0.26, F(2)=0.76, F(3)=0.98, F(4)=1.00 Binomial cumulative distribution function values are tabulated in the back of the course textbook BINOMIAL DISTRIBUTION BINOMIAL DISTRIBUTION For a single trial Probability of success is p Probability of failure is 1-p Use multiplication rule of probability for the trials Example Having 2 successes out of 3 with probability being 0.90 of having a success Success, success, failure Success, failure, success Failure, success, success Total of 3 ways of this happening Have 3 choose 2 ways of getting 2 successes and one failure Probability of getting a single trial is (0.9 ^2) * (0.1^1) (0.9 ^2) * (0.1^1) + (0.9 ^2) * (0.1^1) + (0.9 ^2) * (0.1^1) BINOMIAL PROBABILITY DISTRIBUTION Input parameters x = number of success n = number of trials p = probability of success Have BINOMIAL CUMUATIVE DISTRIBUTION FUNCTION Probability less than or equal to Input parameters x = number of success n = number of trials p = probability of success ENGINEERING EXAMPLE Marketing claim is that 60% of solar-heat installations result in a utility bill reduced by at least one third Probability that this happens in exactly 4 of 5 installations ENGINEERING EXAMPLE Marketing claim is that 60% of solar-heat installations result in a utility bill reduced by at least one third Probability that this happens in at least 4 of 5 installations ANOTHER ENGINEERING EXAMPLE USING TABLE 1 Probability that a column will fail is 0.05, among 16 columns What is the probability that at most two will fail? B(2;16,0.05) Look in table 1 =0.9571 ANOTHER ENGINEERING EXAMPLE USING TABLE 1 Probability that a column will fail is 0.05, among 16 columns What is the probability that at least 4 will fail? Means 4 or 5, ….16 will fail Can make use of the 1-p property of the CDF Look in table 1 1-B(3;16,0.05) 1-.9930=0.9571 LAST ENGINEERING EXAMPLE USING TABLE 1 Probability that 0.2 that any one student will dislike engineering statistics. What is the probability that 5 of 18 students randomly selected students dislike engineering statistics Table 1 b(5:18,0.20) B(5;18,0.20)-B(4;18,0.20) 0.8671-0.7164 0.1507 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 HYPERGEOMETRIC DISTRIBUTION HYPERGEOMETRIC EXAMPLE Shipment of 20 digital voice recorders with 5 defectives If 10 are randomly inspected, what is the probability that 2 of the 10 will be defective MEAN AND VARIANCE OF A DISCRETE PROBABILITY DISTRIBUTION MEAN OF BINOMIAL DISTRIBUTION 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 DISTRIBUTION PARAMETERS POISSON PROBABILITY DISTRIBUTION POISSON PROCESS EXAMPLE Bank receives 6 bad checks a day What is the probability that there will be four bad checks on a given day? 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 POISSON APPROXIMATION EXAMPLE 3,840 generators in the field under warranty Probability is 1/1200 that one will fail in any given year Probability that 1 will fail? Lambda=3840/1200=3.2