Fall Semester 2004

Chapter 5: Probability Distributions

1. What is the purpose of Chapter 5?

2. What is a random phenomenon? Give an example.

3. How does the book define the theoretical probability of an event?

4. How do we quantify random phenomena through observation? What is the relative frequency?

5. What does the law of large numbers state?

6. What is a probability distribution?

7. Give two examples of discrete probability distributions.

8. Why does the book bring up the Poisson distribution?

9. How many parameters does the formula for the Poisson distribution have?

10. Give an example in which the Poisson distribution is usually applicable and explain why.

11. How are discrete distributions usually displayed?

12. How are continuous probability distributions different from the discrete probability distributions? Give an example.

13. What is the probability of any specific value in a continuous probability distribution? (Next page: Why?)

14. What is a PDF and where is it useful?

15. Given a PDF curve how do we calculate the probability associated with a range of values?

16. (Interactive Tutorial) What is the purpose of the first interactive tutorial (on pp. 173-175)?

17. (Interactive Tutorial) What is a uniform distribution?

18. (Interactive Tutorial) Give another example of a situation where a Poisson distribution would be applicable. What are the prerequisites in this situation?

19. (Interactive Tutorial) On the last page of the tutorial, what is the probability of a value being between -0.96 and 0.99 (approximately -1 and 1). How about between 1 and 2.

20. (Interactive Tutorial) Using this last page in the tutorial can you obtain a value of 1 (that is, 100%) for the probability of a value being in a certain range? What would that range be?

21. (Interactive Tutorial) What is the purpose of the two horizontal scroll bars below the chart? What do you use them for?

22. What is a random variable?

23. How is a discrete random variable different from a continuous random variable?

24. Explain the use of upper and lowercase in the context of probability distributions.

25. Define observation, sample, and random sample.

26. In most cases we want our samples to be random samples to give a true picture of the underlying probability distribution. Please consider the following problem:

    During World War II many economists, mathematicians, and statisticians were members of Columbia University's Statistics Research Group, which did high-level consulting work for the armed services. As part of this group's work, statistician Abraham Wald was asked where to place armor on planes. It seemed obvious to the aircraft engineers that armor was needed at the places most frequently hit, as found in a large sample of battle-proven airplanes. After studying the bullet holes of a sample of returning planes, Wald's conclusion was to place the armor where bullet holes were least frequently found in these planes, and that's what he recommended.

    Now the questions:

    1. Was his reasoning justified?
    2. Was there anything wrong with the aircraft engineers' sampling design?
    3. Did they overlook anything?

    Part of the challenge in statistics is to remove all bias from sampling. This is difficult to do and subtle biases can creep into even the most carefully designed studies. Here's another problem:

    ABC's 20/20 television broadcast on July 16, 1993 reported on a study in which individuals who had lived to be 100 years of age or more were queried in the hope of finding common characteristics. The implication was drawn that if a younger person worked at acquiring the characteristics shared by these centenarians, then the probability of reaching such an old age increased. Why was this study design inappropriate for the implication drawn?

27. (Interactive Tutorial) What is the second tutorial about?

28. (Interactive Tutorial) How does the observed distribution compare if you increase the number of shots taken?

29. (Interactive Tutorial) The distribution of the shots around the target is described by a ________ density function because it involves two random variables.

30. (Interactive Tutorial) How is the histogram in the tutorial describing the distribution of shots?

31. What is the normal distribution?

32. How many parameters does the normal distribution have?

33. What are they?

34. (Interactive Tutorial) What is the name of the third interactive tutorial in this chapter?

35. (Interactive Tutorial) Name the distributions presented in the tutorial.

36. What percent of the values in a normal distribution is located within one standard deviation to the right of the mean?

37. What EXCEL functions can you use to work with the normal distribution?

38. What does NORMDIST(40, 50, 4, TRUE) calculate?

39. What does NORMDIST(40, 50, 4, FALSE) calculate?

40. What does NORMINV(0.90, 50, 4) calculate?

41. How do you check if your data is normally distributed?

42. What is a normal score?

43. What is a standard normal distribution?

44. Explain what the normal probability plot is.

45. (Interactive Tutorial) What is the name of the next file you work with?

46. One step is missing in the instructions for the experiment on page 188. What is it?

47. What does the normal probability plot indicate in this case?

48. How do you calculate the expected batting average if you know the normal score?

49. Does the normal probability plot show if the data is skewed or not?

50. What does the title (Parameters and Estimators) of the section that starts on page 191 refer to?

51. Explain what we mean by consistent estimators.

52. Provide an answer to the following question (which appears at the bottom of page 191): "How large must a sample be to estimate accurately the value of mu?"

53. Explain what we mean by the sampling distribution. Whose distribution is it?

54. Explain what the standard error is and how it is related to the sampling distribution.

55. State the Central Limit Theorem.