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## Last Time

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**Last Time**• Histograms • Binomial Probability Distributions • Lists of Numbers • Real Data • Excel Computation • Notions of Center • Average of list of numbers • Weighted Average**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)**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**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**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**Reading In Textbook**Approximate Reading for Today’s Material: Pages 277-282, 34-43 Approximate Reading for Next Class: Pages 55-68, 319-326**Big Picture**• Margin of Error • Choose Sample Size Need better prob tools Start with visualizing probability distributions**Big Picture**• Margin of Error • Choose Sample Size Need better prob tools Start with visualizing probability distributions, Next exploit constant shape property of Bi**Big Picture**Start with visualizing probability distributions, Next exploit constant shape property of Binom’l**Big Picture**Start with visualizing probability distributions, Next exploit constant shape property of Binom’l Centerpoint feels p**Big Picture**Start with visualizing probability distributions, Next exploit constant shape property of Binom’l Centerpoint feels p Spread feels n**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**Notions of Center**Will later study “notions of spread”**Notions of Center**Textbook: Sections 4.4 and 1.2 Recall parallel development: (a) Probability Distributions (b) Lists of Numbers Study 1st, since easier**Notions of Center**• Lists of Numbers “Average” or “Mean” of x1, x2, …, xn Mean = = common notation**Notions of Center**Generalization of Mean: “Weighted Average” Intuition: Corresponds to finding balance point of weights on number line**Notions of Center**Generalization of Mean: “Weighted Average” Intuition: Corresponds to finding balance point of weights on number line**Notions of Center**Textbook: Sections 4.4 and 1.2 Recall parallel development: (a) Probability Distributions (b) Lists of Numbers**Notions of Center**• Probability distributions, f(x) Approach: use connection to lists of numbers**Notions of Center**• Probability distributions, f(x) Approach: use connection to lists of numbers Recall: think about many repeated draws**Notions of Center**• Probability distributions, f(x) Approach: use connection to lists of numbers Draw X1, X2, …, Xn from f(x)**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)**Notions of Center**Rearrange list, depending on values**Notions of Center**Number of Xis that are 1**Notions of Center**Apply Distributive Law of Arithmetic**Notions of Center**Recall “Empirical Probability Function”**Notions of Center**Frequentist approximation**Notions of Center**A weighted average of values that X takes on**Notions of Center**A weighted average of values that X takes on, where weights are probabilities**Notions of Center**A weighted average of values that X takes on, where weights are probabilities This concept deserves its own name: Expected Value**Expected Value**Define Expected Value of a random variable X:**Expected Value**Define Expected Value of a random variable X:**Expected Value**Define Expected Value of a random variable X: Useful shorthand notation**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**Expected Value**E.g. Roll a die, bet (as before):**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**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”**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”**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?**Expected Value**Let X = “net winnings”**Expected Value**Let X = “net winnings” Weighted average, wts & values**Expected Value**Let X = “net winnings” Weighted average, wts & values**Expected Value**Let X = “net winnings” i.e. weight average of values 9, -4 & 0, with weights of “how often expect”, thus “expected”**Expected Value**Let X = “net winnings” Conclusion: on average in many plays, expect to win $1 per play.**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)**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**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