1 / 54

Econ 240A

Econ 240A. Power Three. Summary: Week One. Descriptive Statistics measures of central tendency measures of dispersion Distributions of observation values Histograms: frequency(number) Vs. value Exploratory data Analysis stem and leaf diagram box and whiskers diagram. Probability.

josephine-kirkland
Download Presentation

Econ 240A

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Econ 240A Power Three

  2. Summary: Week One • Descriptive Statistics • measures of central tendency • measures of dispersion • Distributions of observation values • Histograms: frequency(number) Vs. value • Exploratory data Analysis • stem and leaf diagram • box and whiskers diagram

  3. Probability The Gambler Kenny Rogers 20 Great Years

  4. Outline • Why study probability? • Random Experiments and Elementary Outcomes • Notion of a fair game • Properties of probabilities • Combining elementary outcomes into events • probability statements • probability trees

  5. Outline continued • conditional probability • independence of two events

  6. Perspectives About Probability • Logical Discipline (like economics) • Axiomatic: conclusions follow from assumptions • Easier to Understand with Examples • I will use words, symbols and pictures • Test Your Understanding By Working Problems

  7. Why study probability? • Understand the concept behind a random sample and why sampling is important • independence of two or more events • understand a Bernoulli event • example; flipping a coin • understand an experiment or a sequence of independent Bernoulli trials

  8. Cont. • Understand the derivation of the binomial distribution, i.e. the distribution of the number of successes, k, in n Bernoulli trials • understand the normal distribution as a continuous approximation to the discrete binomial • understand the likelihood function, i.e. the probability of a random sample of observations

  9. Uncertainty in Life • Demography • Death rates • Marriage • divorce

  10. Uncertainty in Life: US (CDC)

  11. Probability of First Marriage by Age, Women: US (CDC)

  12. Cohabitation: The Path to Marriage?: US(CDC)

  13. Race/ethnicity Affects Duration of First Marriage

  14. Concepts • Random experiments • Elementary outcomes • example: flipping a coin is a random experiment • the elementary outcomes are heads, tails • example: throwing a die is a random experiment • the elementary outcomes are one, two, three, four, five, six

  15. Axiomatic Basis or Concepts H • Elementary outcomes have non-negative probabilities: P(H)>=0, P(T)>=0 • The sum of the probabilities over all elementary outcomes equals one: P(H) + P(T) = 1 H Flip a coin T

  16. Axiomatic Basis or Concepts II • The probability of two mutually exculsive events is zero: P(H and T) = P(H^T) = 0 • The probability of one outcome or the other is the sum of the probabilities of each minus any double counting: P(H or T) = P(H U T) = P(H) + P(T) – P(H^T) • The probability of the event not happening is one minus the probability of the event happening:

  17. Axiomatic Basis or Concepts III • Conditional probability of heads given tails equals the joint probability divided by the probability of tails: P(H/T) = P(H^T)/P(T)

  18. Concept • A fair game • example: the probability of heads, p(h), equals the probability of tails, p(t): p(h) = p(t) =1/2 • example: the probability of any face of the die is the same, p(one) = p(two) = p(three) = p(four) =p(five) = p(six) = 1/6

  19. Properties of probabilities • Nonnegative • example: p(h) • probabilities of elementary events sum to one • example p(h) + p(t) = 1

  20. Another Example: Toss Two Coins H2 H, H H1 T2 H, T H2 T, H T1 T2 T, T

  21. Flipping a coin twice: 4 elementary outcomes heads h, h heads tails h, t heads t, h tails t, t tails

  22. Axiomatic Basis or Concepts H • Elementary outcomes have non-negative probabilities: P(H, H)>=0, P(H, T)>=0, P(T, H)>=0, P(T, T) >=0 • The sum of the probabilities over all elementary outcomes equals one: P(H, H) + P(H, T) + P(T, H) + P(T, T) = 1 • The probability of two mutually exculsive events is zero: P[(H, H)^(H, T)] = 0

  23. Axiomatic Basis or Concepts II • The probability of one outcome or the other is the sum of the probabilities of each minus any double counting: P[(H, H) U (H,T)] = P(H, H) + P(H, T) – P[(H, H)^(H, T)] = P(H, H) + P(H, T) • The probability of the event not happening is one minus the probability of the event happening:

  24. Axiomatic Basis or Concepts III • Conditional probability of heads, heads given heads, tails equals the joint probability divided by the probability of heads, tails: P[(H, H)/(H, T)] = P[(H, H)^(H, T)]/P(H, T)

  25. Throwing Two Dice, 36 elementary outcomes

  26. Larry Gonick and Woollcott Smith, The Cartoon Guide to Statistics

  27. Combining Elementary Outcomes Into Events • Example: throw two dice: event is white die equals one • example: throw two dice and red die equals one • example: throw two dice and the sum is three

  28. Event: white die equals one is the bottom row Event: red die equals one is the right hand column

  29. Combining Elementary Outcomes Into Events • Example: throw two dice: event is white die equals one P(W1) =P(W1^R1) + P(W1^R2) + P(W1^R3) + P(W1^R4) + P(W1^R5) + P(W1^R6) = 6/36 • example: throw two dice and red die equals one • example: throw two dice and the sum is three

  30. Event: 2 dice sum to three is lower diagonal

  31. Operations on events • The event A and the event B both occur: • Either the event A or the event B occurs or both do: • The event A does not occur, i.e.not A:

  32. Probability statements • Probability of either event A or event B • if the events are mutually exclusive, then • probability of event B

  33. Probability of a white one or a red one: p(W1) + p(R1) double counts

  34. Two dice are thrown: probability of the white die showing one and the red die showing one

  35. Probability 2 dice add to 6 or add to 3 are mutually exclusive events Probability of not rolling snake eyes is easier to calculate as one minus the probability of rolling snake eyes

  36. Problem • What is the probability of rolling at least one six in two rolls of a single die? • At least one six is one or two sixes • easier to calculate the probability of rolling zero sixes: (5/36 + 5/36 + 5/36 + 5/36 + 5/36) = 25/36 • and then calculate the probability of rolling at least one six: 1- 25/36 = 11/36

  37. Probability tree 1 2 1 3 4 2 5 2 rolls of a die: 36 elementary outcomes, of which 11 involve one or more sixes 3 6 4 5 6

  38. Conditional Probability • Example: in rolling two dice, what is the probability of getting a red one given that you rolled a white one? • P(R1/W1) ?

  39. In rolling two dice, what is the probability of getting a red one given that you rolled a white one?

  40. Conditional Probability • Example: in rolling two dice, what is the probability of getting a red one given that you rolled a white one? • P(R1/W1) ?

  41. Independence of two events • p(A/B) = p(A) • i.e. if event A is not conditional on event B • then

  42. Concept • Bernoulli Trial • two outcomes, e.g. success or failure • successive independent trials • probability of success is the same in each trial • Example: flipping a coin multiple times

  43. Problem 6.28 Distribution of a retail store purchases classified by amount and method of payment

  44. Problem (Cont.) • A. What proportion of purchases was paid by debit card? • B. Find the probability a credit card purchase was over $100 • C. Determine the proportion of purchases made by credit card or debit card

  45. Problem 6.28

  46. Problem (Cont.) • A. What proportion of purchases was paid by debit card? 0.36 • B. Find the probability a credit card purchase was over $100 • C. Determine the proportion of purchases made by credit card or debit card

  47. Problem (Cont.) • A. What proportion of purchases was paid by debit card? • B. Find the probability a credit card purchase was over $100 p(>$100/credit card) = 0.23/0.47 = 0.489 • C. Determine the proportion of purchases made by credit card or debit card

  48. Problem (Cont.) • A. What proportion of purchases was paid by debit card? • B. Find the probability a credit card purchase was over $100 • C. Determine the proportion of purchases made by credit card or debit card • note: credit card and debit card purchases are mutually exclusive • p(credit or debit) = p(credit) + p (debit) = 0.47 + 0.36

  49. Problem 6.61 • A survey of middle aged men reveals that 28% of them are balding at the crown of their head. Moreover, it is known that such men have an 18% probability of suffering a heart attack in the next ten years. Men who are not balding in this way have an 11% probability of a heart attack. Find the probability that a middle aged man will suffer a heart attack in the next ten years.

More Related