2. Random Numbers Toss 1 quarter – What is the probability of getting a head? Toss 1 quarter 10 times – What is the proportion of heads you get? Toss 1 quarter 40 times – What is the proportion of heads you get? Excel Exercise 1

3. The outcome of a coin toss The time between customers arriving at an ATM The price of a share of a company’s stock at the close of the market

4. Independence Trials (events, outcomes) must be independent of each other. Probability is empirical Computer simulation is faster, but it’s not empirical

5. Probability Models Create a list of the possible outcomes Assign probabilities to each outcome. Phone book – Benford’s Law

6. First digits of phone number suffixes (e.g., 123- 4 567) First digits of addresses 1 2 3 4 5 6 7 8 9 01 2 3 4 5 6 7 8 9

7. Benford’s Law In many data, the distribution of “first digits” can be modeled as a (-log) distribution. This means that the digit “1” has a much higher probability of occurring than if the digits were uniformly distributed.

10. Probability Models

11. e.g., An “outcome” can a particular random sample of 55,000 households out of 106,000,000 households. Suppose there are five students. If we want to draw a random sample with n=3, we have 10 distinct possibilities for samples And therefore the sample space is ….

12. the sample space is …. Jack Dan JillJack Jill HelenDan Harry Helen Jack Dan HarryJack Harry HelenJill Harry Helen Jack Dan HelenDan Jill Harry Jack Jill HarryDan Jill Helen

13. Jack Dan JillJack Jill HelenDan Harry Helen Jack Dan HarryJack Harry HelenJill Harry Helen Jack Dan HelenDan Jill Harry Jack Jill HarryDan Jill Helen Each of these is an event

14. Jack Dan Jill10%4%5%9%3% Jack Dan Harry10%12% 5%9% Jack Dan Helen10%7%13%3%25% Jack Jill Harry10%0%4%8%24% Jack Jill Helen10%7%10%2%20% Jack Harry Helen10%16%14%16%7% Dan Jill Harry10%16%12%15%2% Dan Jill Helen10% 11%6% Dan Harry Helen10%19%14%17%0% Jill Harry Helen10%8%5%14%5% sum100%

15. What if we only care about the sum of the dots, not their order nor how we get to the sum? How many alternatives are there? What if we only care about the number of dots on the dice, but not their order? How many alternatives are there?

16. A s an event

20. Assigning Probabilities: Intervals of Outcomes There are 10 possible events. What is P(0.2<X<0.6)

21. Assigning Probabilities: Intervals of Outcomes There are 50 possible events. What is P(0.2<X<0.6)

22. Assigning Probabilities: Intervals of Outcomes There are 5000 possible events. What is P(0.2<X<0.6)

23. This Normal distribution is the idealized equivalent of a Normal probability model.

24. This Normal probability model (empirical) is idealized by the Normal distribution (computational).

26. Random Variables

27. Icosahedron X Y X+Y

28. Assigning Probabilities: Intervals of Outcomes Both are uniformly distributed between 0 and 1, but one is a discrete random variable and the other one is continuous.

30. For discrete random variables, the difference between > and ≥ matters.

35. Take the list of integers between 0 to 50. The probability of randomly picking any individual integer out of that sample space, say, 23, is 1/50, one fiftieth. n5050050,00050million50 trillion -> Infinity P(X=23)1/501/5001/50,0001/50,000, 000 1/50,000, 000,000, 000 -> 0

36. For continuous random variables, the difference between > and ≥ doesn’t matter because the probability of picking any one number is basically zero.

37. If I buy a lottery ticket for $1 but win nothing, my net gain is -$1. If I win (and the probability is 0.00001) $100,000, I my net gain is $99,999 Then on average, my net gain from playing the lottery is Wrong!

38. If I buy a lottery ticket for $1 but win nothing, my net gain is -$1. If I win (and the probability is 0.00001) $100,000, I my net gain is $99,999 Then on average, my net gain from playing the lottery is This is the expected value of X

39. Excel Exercise 2

42. Excel Exercise 3

43.  =0 Not independent

45. The Sampling Distribution of a Sample Mean

46. Hershey Kisses: if sample size n equals 5, what are the possible values for x, the number of silver kisses?

47. Suppose there are five students. If we want to draw a random sample with n=3, we have 10 distinct possibilities for samples Jack Dan JillJack Jill HelenDan Harry Helen Jack Dan HarryJack Harry HelenJill Harry Helen Jack Dan HelenDan Jill Harry Jack Jill HarryDan Jill Helen

48. Suppose there are 1000 individuals. We can collect information from small samples (n=30, n=60, n=100, n=300). What will be the sampling distribution of the mean (the distribution of the sample mean)?

49. kernel density function of 1000 normal random numbers

50. kernel density function of normal random numbers in samples n = 30

51. kernel density function of normal random numbers in samples n = 60

52. kernel density function of normal random numbers in samples n = 100

53. kernel density function of normal random numbers in samples n = 300

54. kernel density function of 1000 normal random numbers

55. PopulationSample 1Sample 2Sample 3 mean 201,209 174,898 145,792 221,358 st dev 199,964 203,879 176,502 200,407 Sample 4Sample 5Sample 6Sample 7 mean 216,957 205,055 189,929 245,484 st dev 209,431 247,535 217,423 177,200 Sample 8Sample 9Sample 10 mean 207,733 231,096 201,115 st dev 207,355 189,594 170,539 n=60

56. hist x in 1/60 hist x in 61/120 hist x in 121/180 hist x in 181/240

57. . sum x in 1/60 Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- x | 60 -.0297738.945766 -1.787513 1.953314. sum x in 61/120 Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- x | 60 -.0680528.9385375 -2.528759 1.983905. sum x in 121/180 Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- x | 60 -.1058176 1.042179 -2.375701 2.154691. sum x in 181/240 Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- x | 60.1628009.8848563 -1.823286 1.794983

59. Excel Exercise 4

60.  = 0.0068 n=3 n=30 n=60 n=100

61. n=30meanstandard error 30.11440.4978 30-0.00010.1752 60-0.04210.1249 100-0.0110.0967 population0.0068n.a.

64. Notice that the Central Limit Theorem doesn’t say “Draw an SRS of size n from a Normally distributed population.” It says “any” population. Excel Exercise 5 and 6