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Probability Theory

Probability Theory. http://stats.stackexchange.com/questions/ 423/what-is-your-favorite-data-analysis-cartoon. Example: Independence. Are the following events independent or dependent? Winning at the Hoosier (or any other) lottery.

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Probability Theory

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  1. Probability Theory http://stats.stackexchange.com/questions/ 423/what-is-your-favorite-data-analysis-cartoon

  2. Example: Independence Are the following events independent or dependent? • Winning at the Hoosier (or any other) lottery. • The marching band is holding a raffle at a football game with two prizes. After the first ticket is pulled out and the winner determined, the ticket is taped to the prize. The next ticket is pulled out to determine the winner of the second prize.

  3. Example: Independence • Deal two cards without replacement A = 1st card is a heart B = 2nd card is a heart C = 2nd card is a club. • Are A and B independent? • Are A and C independent? 2. Repeat 1) with replacement.

  4. Disjoint vs. Independent In each situation, are the following two events a) disjoint and/or b) independent? • Draw 1 card from a deck A = card is a heart B = card is not a heart • Toss 2 coins A = Coin 1 is a head B = Coin 2 is a head • Roll two 4-sided dice. A = red die is 2 B = sum of the dice is 3

  5. Example: Complex Multiplication Rule (1) The following circuit is in a series. The current will flow only if all of the lights work. Whether a light works is independent of all of the other lights. If the probability that A will work is 0.8, P(B) = 0.85 and P(C) = 0.95, what is the probability that the current will flow? A B C http://www.berkeleypoint.com/learning/parallel_circuit.html

  6. Example: Complex Multiplication Rule (2) The following circuit to the right is parallel. The current will flow if at least one of the lights work. Whether a light works is independent of all of the other lights. If the probability that A will work is 0.8, P(B) = 0.85 and P(C) = 0.95, what is the probability that the current will flow? A B C http://www.berkeleypoint.com/learning/parallel_circuit.html

  7. Example: Complex Multiplication Rule (3) A diagnostic test for a certain disease has a specificity of 95%. The specificity is the same as true negative, that is the test is negative when the person doesn’t have the disease. a) What is the probability that one person has a false positive (the test is positive when they don’t have the disease)? b) What is the probability that there is at least one false positive when 50 people who don’t have the disease are tested?

  8. Example: General Addition Rule (1) Select a card at random from a deck of cards. What is the probability that the card is either an Ace or a Heart?

  9. Example: General Addition Rule (2) At a certain University, the probability that a student is a math major is 0.25 and the probability that a student is a computer science major is 0.31. In addition, the probability that a student is a math major and a student science major is 0.15. a) What is the probability that a student is a math major or a computer science major? b) What is the probability that a student is a computer science major but is NOT a math major?

  10. Probability Theory http://stats.stackexchange.com/questions/ 423/what-is-your-favorite-data-analysis-cartoon

  11. Conditional Probability: Example A news magazine publishes three columns entitled "Art" (A), "Books" (B) and "Cinema" (C). Reading habits of a randomly selected reader with respect to these columns are a) What is the probability that a reader reads the Art column given that they also read the Books column? b) What is the probability that a reader reads the Books column given that they also read the Art column?

  12. Example: General Multiplication Rule Suppose that 8 good and 2 defective fuses have been mixed up. To find the defective fuses we need to test them one-by-one, at random. Once we test a fuse, we set it aside. a) What is the probability that we find both of the defective fuses in the first two tests? b) What is the probability that the first tested fuse is good and the last two tested are defective?

  13. Example: Tree Diagram/Bayes’s Rule A diagnostic test for a certain disease has a 99% sensitivity and a 95% specificity. Only 1% of the population has the disease in question. If the diagnostic test reports that a person chosen at random from the population tests positive, what is the probability that the person does, in fact, have the disease?

  14. Binomial Setting Do the following use the Binomial Setting? • Rolling a fair 4-sided die five times and observing whether the number showing is a 1 or not • In a drug trial, 20 patients with the same condition are given a drug and some are given a placebo to see if the drug is effective or not. • In quality control we want to see if a particular product is ‘not acceptable’. We take 20 random samples from an assembly line that uses different machines to produce the product.

  15. Examples of Binomial Distribution • In a clinical trial, a patient’s condition may improve or not. We study the number of patients who improved. • Was a sales transaction considered pleasant? The binomial distribution describes the number of pleasant transactions. • In quality control we assess the number of defective items in a lot of goods.

  16. Example: Binomial Distribution Suppose 20% of all copies of a particular textbook fail a certain binding strength test. Let's check a batch of 15 such textbooks. • Is this a binomial distribution? • What is the chance that we get no defective textbooks? • What is the chance that we get less than 3 defective textbooks? • What is the chance that we get more than 2 defective textbooks?

  17. Example: Binomial Distribution (cont)

  18. Histograms of Binomial Distributions n = 10 p = 0.5 n = 10 p = 0.25 n = 10 p = 0.75

  19. Example: Binomial Distribution (cont) Suppose 20% of all copies of a particular textbook fail a certain binding strength test. Let's check a batch of 15 such textbooks. • What are the mean and variance of the number of textbooks that will fail the binding test?

  20. Example: Normal Approximation to the Binomial The ideal size of a first-year class at a particular college is 150 students. The college, knowing from past experience that on the average only 30 percent of those accepted for admission will actually attend, uses a policy of approving the applications of 450 students. Compute the probability that more than 150 students attend this college.

  21. Example: Poisson Distribution An IT consultant receives an average of 3 calls per hour. Let X be the number of calls the consultant receives next hour. X follows a Poisson distribution. a) What is the chance that the consultant receives exactly one call during the next hour? b) What is the chance that the consultant receives more than one call during the next hour?

  22. Example: Poisson Approximation to Binomial 0.2% of feral cats are infected with feline aids (FIV) in a region. What is the chance that there are exactly 10 cats infected with FIV among 1000 cats?

  23. Example: Uniform A person casually walks to the bus stop when the bus comes every 30 minutes has a density function of • What is the proportion of people that have to wait between 5 and 10 minutes? • What is the median of people who wait? • What is the mean of people who wait?

  24. Example: Uniform (2) The density function of the uniform distribution over the interval [a,b] is

  25. Example: Uniform (3) A packaging line constantly packages 200 cartons per hour. After weighing every package variation the distribution of the weights was found to be uniform with weights ranging from 18.2 lbs. – 20.4 lbs., measured to the nearest tenths. The customer requires less than 20.0 lbs. for ergonomic reasons. What is the probability that the package weights less than 20 lbs.?

  26. Example: Exponential The life span of some bacteria (in hours) has an exponential distribution with parameter 2. • What is the proportion of bacteria that live at most 1 hour? • What is the proportion of bacterial that live more than 1.5 hours? • What are the mean and standard deviation of the distribution of these bacteria?

  27. Gamma Distribution • Generalization of the exponential function • Uses • probability theory • theoretical statistics • actuarial science • operations research • engineering

  28. Beta Distribution • This distribution is only defined on an interval • standard beta is on the interval [0,1] • uses • modeling proportions • percentages • probabilities

  29. Other Continuous Random Variables • Weibull • exponential is a member of family • uses: lifetimes • lognormal • log of the normal distribution • uses: products of distributions • Cauchy • symmetrical, long straggly tails

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