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  1. Module 2: Probability, Random Variables & Probability DistributionsModule 2b T.J. Harris

  2. Random Variable What is a random variable? • When experiments lead to categorical results, we assign numbers to the random variable: • e.g., defective = 0, functional = 1 Why do we assign numbers? • to help us to express probabilities and outcomes using mathematical expressions T.J. Harris

  3. Types of Random Variables Discrete Random Variables • take on integer values or other discrete sets of values Continuous Random Variables • take on values from a portion of the real line T.J. Harris

  4. Random Variables - Notation Random variable denoted by capital X Particular values obtained from an experiment are denoted by lower-case x T.J. Harris

  5. Discrete Random Variables • Notation for probability functions • Example - sampling one chip from a batch of 30, when 10 are defective. • defective = 0, functional = 1 T.J. Harris

  6. Cumulative Distribution Function We can also use Cumulative Distribution Functions • FX is the probability that we obtain an outcome less than or equal to a given number T.J. Harris

  7. Galvanneal Line - Revisited Assign discrete random variable (number) to reflect outcomes x=0, 1, 2 -- “acceptability score” • O1: thickness off-spec, fails tape test x = 0 • O2: thickness acceptable, fails tape test x = 1 • O3: thickness off-spec, passes tape test x = 2 • O4: thickness acceptable, passes tape test x = 3 T.J. Harris

  8. Galvanneal Line - Revisited • P(x=0) = 0.04 P(x=1) = 0.10 • P(x=2) = 0.03 P(x=3) = 0.83 T.J. Harris

  9. Expected Value • What is the value of the random variable on average? • Let’s figure out the expected value of the outcome for testing of a sample from the Galvannealline. • Maybe it will help to consider what would happen if we tested 1000 random samples T.J. Harris

  10. Expected Value The expected value of a discrete random variable X is defined as The expected value is given the symbol : • is called the MEAN of the random variable X. What is the expected number of buttons if we selected someone randomly from the class? T.J. Harris

  11. Example - Mean for Galvanneal Line • Using the probability function, What is E(X) when we are rolling a die? T.J. Harris

  12. Expected Values In general, if we have any function of a random variable, we can find the expected value of that function: • What is E(X3) when we roll a die? • Suppose we ‘pay out’ the following amounts for each roll of the die: (1, $1), (2, $6),(3,$4),(4,$1),(5,$0),(6,$ 1) • What is the expected ‘pay out’ from a roll of the die? T.J. Harris

  13. Variance What is the expected squared deviation from the mean? • The sample variance s2 which is a statistic used to estimate the true variance 2 • What is 2 when we roll a die? • What is 2 for the number of buttons? T.J. Harris

  14. Standard Deviation … is the square root of the variance The mean, variance and standard deviation are parameters summarizing a probability distribution for a random variable. T.J. Harris

  15. Linearity of Expectation The Expected Value operation is LINEAR: • Additivity E(X+Y) = E(X) + E(Y) 2) Scaling E(kX) = k E(X) where k is a constant e.g., E(7X+6) = 7E(X) + 6 = 7X + 6 T.J. Harris

  16. Probability Distributions for Discrete R.V.’s • We can determine probability functions by counting • Some common situations result in • Binomial Distribution • Poisson Distribution T.J. Harris

  17. Binomial Distribution Suppose we are conducting a number of independent trials, each with only one of two possible values • Each trial is called a Bernoulli trial • Outcomes -- 0, 1 -- True/False -- Success/Fail -- ... • For each trial, P(1) = p, and P(0) = 1-p • If we have n trials, what is the probability that we obtain x successes (outcomes of 1)? T.J. Harris

  18. Binomial Distribution The probability of having x successes in n independent trials is: Binomial Probability Distribution Function T.J. Harris

  19. Binomial Distribution Mean Variance How could we prove this? T.J. Harris

  20. T.J. Harris

  21. Using the Binomial Distribution Sampling with Replacement - Example - On the microwave module line of a telecommunications equipment maker, the probability of a defective module is 0.21. From each batch, one module is selected and tested, and then returned to the batch. This procedure is repeated 5 times, so that we have 5 independent tests for defects. What is the probability of having : a) 1 defect in the five tests? b) 3 defects in the five tests? Why is it important that the module be returned? Would anyone really do this? T.J. Harris

  22. Binomial Example a) n = 5 (independent trials), x = 1 (“success” = defect identified) b) n = 5, x = 3 T.J. Harris

  23. Binomial Example Why is it necessary to return the module to the batch before the next sample? • preserve independence • if module not returned, chance of getting a defective module changes. • Binomial distribution is appropriate in sampling situations when there is “sampling with replacement” • for sampling without replacement, we need to use the Hypergeometric distribution • if the batch size is large, relative to the number of tests in the sample, binomial provides reasonable approximation • e.g., sampling 10 items from a population of 10000 T.J. Harris

  24. Examples (Or Not) of Appropriateness of Binomial Distributions (3-87) A production process produces thousands of temperature transducers. Let X denote the number of nonconforming transducers in a sample of size 30 selected at random from the process. Let X denote the number of express mail packages received by the post office in a 24-hr period. Forty randomly selected semiconductor chips are tested. Let X denote the number of chips in which the test finds at least one contamination particle. T.J. Harris

  25. Examples (Or Not) of Appropriateness of Binomial Distributions (3-87) A filling operation attempts to fill detergent packages to the advertised weight. Let X denote the number of detergent packages that are under filled. Errors in a digital communication channel occur in bursts that affect several consecutive bits. Let X denote the number of bits in error in a transmission of 100,000 bits. Let X denote the number of correct answers by a student taking a multiple choice exam in which a student can eliminate some of the choices as being incorrect in some questions and all of the incorrect choices in other questions. T.J. Harris

  26. Overbooking (3-97 & extension) • Because not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds 120 passengers. The probability that a passenger does not show up is 0.10. The passengers act independently of each other. • What is the probability that every passenger that shows up gets a seat? • What is the probability that the flights departs with empty seats? • What are the mean & standard deviation for flight occupancy? T.J. Harris

  27. Overbooking Let X be the number of passengers with tickets that do not show up. X is Binomial with n = 125 and p = 0.1 A B T.J. Harris

  28. Overbooking T.J. Harris

  29. Overbooking E(X) = np = 125(.1) = 12.5, E(n-X) = n-E(X) = 112.5 T.J. Harris

  30. Overbooking This type of analysis shows that a cost only approach can be used to justify an organization's poor customer service or failure to comply with environmental & safety regulations T.J. Harris