Chapter 6 Lecture Notes

For errata, see Corrections to Sullivan’s Fundamentals of Statistics.

Looking ahead to sec 7.4, you will need MATH200A Program part 5.

6.1  Discrete Random Variables

284 Last chapter we briefly met probability models; here we look at them more closely.

(review definition of probability as long-term rel. freq.)

285 notation: X is the variable, x is a particular value (data point)

285 distinguish discrete and continuous random variables: only certain points on the number line versus all values (perhaps limited to a range)

286 discrete probability distribution = probability model from last chapter

288 prob. histogram = rel. freq. histogram (Use MATH200A part 1 as before)

probability represented by area

289 mean of DPD need not be a possible value of X

interpret mean value as a mean outcome of a zillion trials

why is it μ and not x̄?

use 1-VarStats rather than formula; check for n=1 (verify this against book example)

291 mean = expected value

not in book: fair price of a game = cost of a ticket such that expected value is 0, meaning everyone breaks even in the long run

Computing: Fair Price = Expected Value (to consumer) + Cost

Example: Insureco charges $200 for a policy. In the policy year, on average, one of every 10,000 policies pays a claim of $400,000, one of every 1000 pays a claim of $60,000, one of every 50 pays a claim of $4,000, and the rest pay no claims. What is the expected value of a policy (to the insurance company), and what is the fair price? (adapted from John Allen Paulos, A Mathematician Plays the Stock Market)

Answer: EV = $20; company’s gross profit is $20 per policy on average. To you, EV = −$20 because consumers lose $20 per policy on average.

FP = −$20 + $200 = $180

292 s.d. of a DPD — compare to s.d. of a rel. freq. dist.

note: σ not s

again, compute using 1-VarStats

interpretation (not in book) — variability, to consider in relation to size of the mean

Example: should you park in a lot or on the street? If you park in a lot, it’s $10 for less than an hour (p = 25%) and $14 for more than an hour. If you park on the street, you might receive a simple $30 parking ticket (p = 20%), or a $100 citation for obstruction of traffic (p = 5%), but of course you might get neither (p = 75%). Which should you do? (adapted from John Allen Paulos, A Mathematician Plays the Stock Market)

Answers: Parking in a lot has μ = $13, σ = $1.73. Parking on the street has μ = $11, σ = $23.64. Street parking represents a slightly lower expected value (average cost), but very much greater uncertainty. Now, do you feel lucky?

293 try review #1–6

6.2  Binomial Probability Distribution

298 know criteria for binomial experiment (criteria 2 and 4 are the same)

typo in box at bottom: X should be x

300 success and failure have special meanings

302 Briefly look at formula

Use MATH200A part 3 for calculations.

try Example 3 using calculator

305 You need the formulas at bottom. Yes, if n≤46 you can get them from the histogram, MATH200A part 4, but if n≥47 the program can’t draw the histogram.

306 Use MATH200A part 4 to make histograms.

Example: 10-question true/false: what are mean and s.d.? Implications?

308 know rule of thumb for normal approximation: VAR(x) = np(1−p) ≥ 10

Use Empirical Rule (when it applies) to identify unusual results. Example 8 is preview of inferential stats, so don’t spend too much time on it at this stage.

310 try review #1,3–6

309 Can you see the flaw in the book’s reasoning in “Should we convict?”

Swain v. Alabama from Web page

What’s New

• 20 Jun 2010: compute fair price only by EV+Cost, not by 1-VarStats of partial data; rewrite pg 308 note to focus on using Empirical Rule (when possible) to identify unusual result
• (intervening changes suppressed
• 4 mar 2009: new document