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Last Time

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  1. Last Time • Histograms • Binomial Probability Distributions • Lists of Numbers • Real Data • Excel Computation • Notions of Center • Average of list of numbers • Weighted Average

  2. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Excel notation to avoid actual calculation • So no computers or calculators • Bring sheet of formulas, etc. • No blue books needed (will just write on my printed version)

  3. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Material Covered: HW 1 – HW 5 • Note: due Thursday, Feb. 19 • Will ask grader to return Mon. Feb. 23 • Can pickup in my office (Hanes 352) • So this weeks HW not included

  4. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Extra Office Hours: • Monday, Feb. 23 8:00 – 9:00 • Monday, Feb. 23 9:00 – 10:00 • Monday, Feb. 23 10:00 – 11:00 • Tuesday, Feb. 24 8:00 – 9:00 • Tuesday, Feb. 24 9:00 – 10:00 • Tuesday, Feb. 24 1:00 – 2:00

  5. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • How to study: • Rework HW problems • Since problems come from there • Actually do, not “just look over” • In random order (as on exam) • Print HW sheets, use as a checklist • Work Practice Exam • Posted in Blackboard “Course Information” Area

  6. Reading In Textbook Approximate Reading for Today’s Material: Pages 277-282, 34-43 Approximate Reading for Next Class: Pages 55-68, 319-326

  7. Big Picture • Margin of Error • Choose Sample Size Need better prob tools Start with visualizing probability distributions

  8. Big Picture • Margin of Error • Choose Sample Size Need better prob tools Start with visualizing probability distributions, Next exploit constant shape property of Bi

  9. Big Picture Start with visualizing probability distributions, Next exploit constant shape property of Binom’l

  10. Big Picture Start with visualizing probability distributions, Next exploit constant shape property of Binom’l Centerpoint feels p

  11. Big Picture Start with visualizing probability distributions, Next exploit constant shape property of Binom’l Centerpoint feels p Spread feels n

  12. Big Picture Start with visualizing probability distributions, Next exploit constant shape property of Binom’l Centerpoint feels p Spread feels n Now quantify these ideas, to put them to work

  13. Notions of Center Will later study “notions of spread”

  14. Notions of Center Textbook: Sections 4.4 and 1.2 Recall parallel development: (a) Probability Distributions (b) Lists of Numbers Study 1st, since easier

  15. Notions of Center • Lists of Numbers “Average” or “Mean” of x1, x2, …, xn Mean = = common notation

  16. Notions of Center Generalization of Mean: “Weighted Average” Intuition: Corresponds to finding balance point of weights on number line

  17. Notions of Center Generalization of Mean: “Weighted Average” Intuition: Corresponds to finding balance point of weights on number line

  18. Notions of Center Textbook: Sections 4.4 and 1.2 Recall parallel development: (a) Probability Distributions (b) Lists of Numbers

  19. Notions of Center • Probability distributions, f(x) Approach: use connection to lists of numbers

  20. Notions of Center • Probability distributions, f(x) Approach: use connection to lists of numbers Recall: think about many repeated draws

  21. Notions of Center • Probability distributions, f(x) Approach: use connection to lists of numbers Draw X1, X2, …, Xn from f(x)

  22. Notions of Center • Probability distributions, f(x) Approach: use connection to lists of numbers Draw X1, X2, …, Xn from f(x) Compute and express in terms of f(x)

  23. Notions of Center

  24. Notions of Center Rearrange list, depending on values

  25. Notions of Center Number of Xis that are 1

  26. Notions of Center Apply Distributive Law of Arithmetic

  27. Notions of Center Recall “Empirical Probability Function”

  28. Notions of Center

  29. Notions of Center Frequentist approximation

  30. Notions of Center

  31. Notions of Center A weighted average of values that X takes on

  32. Notions of Center A weighted average of values that X takes on, where weights are probabilities

  33. Notions of Center A weighted average of values that X takes on, where weights are probabilities This concept deserves its own name: Expected Value

  34. Expected Value Define Expected Value of a random variable X:

  35. Expected Value Define Expected Value of a random variable X:

  36. Expected Value Define Expected Value of a random variable X: Useful shorthand notation

  37. Expected Value Define Expected Value of a random variable X: Recall f(x) = 0, for most x, so sum only operates for values X takes on

  38. Expected Value E.g. Roll a die, bet (as before):

  39. Expected Value E.g. Roll a die, bet (as before): Win $9 if 5 or 6, Pay $4, if 1, 2 or 3, otherwise (4) break even

  40. Expected Value E.g. Roll a die, bet (as before): Win $9 if 5 or 6, Pay $4, if 1, 2 or 3, otherwise (4) break even Let X = “net winnings”

  41. Expected Value E.g. Roll a die, bet (as before): Win $9 if 5 or 6, Pay $4, if 1, 2 or 3, otherwise (4) break even Let X = “net winnings”

  42. Expected Value E.g. Roll a die, bet (as before): Win $9 if 5 or 6, Pay $4, if 1, 2 or 3, otherwise (4) break even Let X = “net winnings” Are you keen to play?

  43. Expected Value Let X = “net winnings”

  44. Expected Value Let X = “net winnings” Weighted average, wts & values

  45. Expected Value Let X = “net winnings” Weighted average, wts & values

  46. Expected Value Let X = “net winnings” i.e. weight average of values 9, -4 & 0, with weights of “how often expect”, thus “expected”

  47. Expected Value Let X = “net winnings” Conclusion: on average in many plays, expect to win $1 per play.

  48. Expected Value Caution: “Expected value” is not what is expected on one play (which is either 9, -4 or 0) But instead on average, over many plays HW: 4.73, 4.74 (1.9, 1)

  49. Expected Value Real life applications of expected value: • Decision Theory • Operations Research • Rational basis for making business decisions • In presence of uncertainty • Common Goal: maximize expected profits • Gives good average results over long run

  50. Expected Value Real life applications of expected value: • Decision Theory • Casino Gambling • Casino offers games with + expected value (+ from their perspective) • Their goal: good overall average performance • Expected Value is a useful tool for this