Question

Choose a course that you are currently taking in which the final exam is worth 100 points. Treating your score on the exam as if it were a continuous uncertain quantity, assess the subjective probability distribution for your score. After you have finished, check your assessed distribution for consistency by:

a. Choosing any two intervals you have judged to have equal probability content, and

b. Determining whether you would be willing to place small evenodds bets that your score would fall in one of the two intervals. (The bet would be called off if the score fell elsewhere.)

c. After assessing the continuous distribution, construct a three-point approximation to this distribution with the extended Pearson-Tukey method. Use the approximation to estimate your expected exam score.

d. Now construct a 5 point approximation with bracket medians. Use this approximation to estimate your expected exam score. How does your answer compare with the estimate from part c?

Answer 1

In part (a),

                    This question requires students to make personal judgments. As an example, suppose the following assessments are made:

P(S ≤ 65) = 0.05

P(S > 99) = 0.05

P(S ≤ 78) = 0.25

P(S ≤ 85) = 0.50

P(S ≤ 96) = 0.75

In part (b),

                  Now the student would ask whether she would be willing to place a 50-50 bet in which she wins if 78 < S ≤ 85 and loses if 85 < S ≤ 96. Is there a problem with betting on an event over which you have some control?

This problem calls for the subjective assessment of odds. Unfortunately, no formal method for assessing odds directly has been provided . Such a formal approach could be constructed in terms of bets or lotteries as in the case of probabilities, but with the uncertainty stated in odds form.

In part (c),

                  Three-point Pearson- Tukey approximation:

Pearson‐Tukey three‐point approximation is measured in units of standard deviation and compared with that of Monte Carlo simulation. Using a variety of well‐known distributions, comparisons are made for the mean of a random variable and for common functions of one and two random variables. Comparisons are also made for the mean of an assortment of risk‐analysis (Monte Carlo) models drawn from the literature. The results suggest that the Pearson‐Tukey approximation is a useful alternative to simulation in risk‐analysis situations.

EP-T(Score) ≈ 0.185 (65) + 0.63 (85) + 0.185 (99) = 83.89.

In part (d),

                  Five-point approximation using bracket medians:

EBM(Score) ≈ 0.2 (68) + 0.2 (80) + 0.2 (85) + 0.2 (93) + 0.2 (98) = 84.8.

Discretization is a common decision analysis technique, of which many methods are described in the literature. The accuracy of these methods is usually judged by how well they match the mean, variance, and possibly higher moments of the underlying distribution.

Hence, 5 point approximation with bracket medians has better mean score than three-point approximation to this distribution .