AGENDA Probability Distributions Random variable concepts A few probability distributions RANDOM VARIABLES CONCEPTS Random variables A function that assigns a value to a possible outcome Discrete – take on specific values in a range Continuous – take on any values in a range Probability distribution… Random variables values can be observed List of possible values and probability of observing each Cumulative Distribution Function… Based on the probability distribution Probability a random variable is less than or = x PROBABILITY DISTRIBUTION AKA probability density List of possible values and probability f(x)=…(usually some equation with input values) 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 Some cumulative distribution function values are tabulated in the back of the course textbook A FEW PROBABILITY DISTRIBUTIONS Binomial… Hypergeometric… Poisson… Many others 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 Computer locking up while surfing the internet Probability of a reboot required is 0.1 for a half hour session BINOMIAL DISTRIBUTION Success = no reboot required= 0.9 Failure = reboot required = 0.1 For 3 different 30 minute sessions No successes, x=0, = FFF One success, x=1, = FFS, FSF, SFF Two successes, x=2, = FSS, SSF, SSF Three successes, x=3, = SSS BINOMIAL DISTRIBUTION Probabilities X=0, 0.1 x 0.1 x 0.1 X=1, 3 times 0.9 x 0.1 x 0.1 X=2, 3 times 0.9 x 0.9 x 0.1 X=3, 0.9 x 0.9 x 0.9 Same as X=0, combination n=3, r= 0 x (0.9^0 x 0.1^3) X=1, combination n=3, r=1 x (0.9^1 x 0.1^2) X=2, combination n=3, r=2 x (0.9^2 x 0.1^1) X=3, combination n=3, r=3 x (0.9^3 x 0.1^0) We can devise a formula for this BINOMIAL PROBABILITY DISTRIBUTION Input parameters x = number of success n = number of trials p = probability of success Have BINOMIAL CUMUATIVE DISTRIBUTION FUNCTION Total probability less than or equal to getting some x 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? Means that 1 or 2 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=0.0070 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 Once you make an observation of an item, you discard the item (without replacement) This changes the probability of each subsequent observation Specify a lot of size = N with some number of successes = 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 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 production machinery Poisson distribution Distribution of number of observations of these “random” events over a particular period of time POISSON DISTRIBUTION PARAMETERS POISSON PROBABILITY DISTRIBUTION POISSON PROCESS EXAMPLE Bank receives 6 bad checks a day on the average 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 Tables may not be of help either Can approximate binomial probabilities with a Poisson approximation Probabilities successes (defectives) is “small” Lots n are “large” N>=20 and p<=0.05 N>=100 and np<=10 Average number of success 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 * 1/1200=3.2