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Chapter 12 The Normal Probability Model

Chapter 12 The Normal Probability Model. 12.1 Normal Random Variable. Black Monday (October, 1987) prompted investors to consider insurance against another “accident” in the stock market. How much should an investor expect to pay for this insurance?

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Chapter 12 The Normal Probability Model

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  1. Chapter 12The Normal Probability Model

  2. 12.1 Normal Random Variable • Black Monday (October, 1987) prompted • investors to consider insurance against • another “accident” in the stock market. How • much should an investor expect to pay for • this insurance? • Insurance costs call for a random variable that can represent a continuum of values (not counts)

  3. 12.1 Normal Random Variable • Prices for One-Carat Diamonds

  4. 12.1 Normal Random Variable • Percentage Change in Stock Market

  5. 12.1 Normal Random Variable • X-ray Measurements of Bone Density

  6. 12.1 Normal Random Variable • With the exception of Black Monday, the histogram of market changes is bell-shaped • Histograms of diamond prices and bone density measurements are bell-shaped • All three involve a continuous range of values; all three can be modeled using normal random variables

  7. 12.1 Normal Random Variable • Definition • A continuous random variable whose • probability distribution defines a standard • bell-shaped curve.

  8. 12.1 Normal Random Variable • Central Limit Theorem • The probability distribution of a sum of • independent random variables of comparable • variance tends to a normal distribution as the • number of summed random variables • increases.

  9. 12.1 Normal Random Variable • Central Limit Theorem Illustrated

  10. 12.1 Normal Random Variable • Central Limit Theorem • Explains why bell-shaped distributions are so common • Observed data are often the accumulation of many small factors (e.g., the value of the stock market depends on many investors)

  11. 12.1 Normal Random Variable • Normal Probability Distribution • Defined by the parameters µ and σ2 • The mean µ locates the center • The variance σ2 controls the spread

  12. 12.1 Normal Random Variable • Standard Normal Distribution (µ = 0; σ2 = 1)

  13. 12.1 Normal Random Variable • Normal Probability Distribution • A normal random variable is continuous and can assume any value in an interval • Probability of an interval is area under the distribution over that interval (note: total area under the probability distribution is 1)

  14. 12.1 Normal Random Variable • Probabilities are Areas Under the Curve

  15. 12.1 Normal Random Variable • Normal Distributions with Different µ’s

  16. 12.1 Normal Random Variable • Normal Distributions with Different σ2’s

  17. 12.2 The Normal Model • Definition • A model in which a normal random variable is • used to describe an observable random • process with µ set to the mean of the data • and σ set to s.

  18. 12.2 The Normal Model • Normal Model for Diamond Prices • Set µ = $4,066 and σ = $738.

  19. 12.2 The Normal Model • Normal Model for Stock Market Changes • Set µ = 0.94% and σ = 4.32%.

  20. 12.2 The Normal Model • Normal Model for Bone Density Scores • Set µ = -1.53 and σ = 1.3.

  21. 12.2 The Normal Model • Standardizing to Find Normal Probabilities • Start by converting x into a z-score

  22. 12.2 The Normal Model • Standardizing Example: Diamond Prices • Normal with µ = $4,066 and σ = $738 • Want P(X > $5,000)

  23. 12.2 The Normal Model • The Empirical Rule, Revisited

  24. 4M Example 12.1: SATS AND NORMALITY • Motivation • Math SAT scores are normally distributed • with a mean of 500 and standard deviation of • 100. What is the probability of a company • hiring someone with a math SAT score of • 600 or more?

  25. 4M Example 12.1: SATS AND NORMALITY • Method – Use the Normal Model

  26. 4M Example 12.1: SATS AND NORMALITY • Mechanics • A math SAT score of 600 is equivalent z = 1. • Using the empirical rule, we find that 15.85% • of test takers score 600 or better.

  27. 4M Example 12.1: SATS AND NORMALITY • Message • About one-sixth of those who take the math • SAT score 600 or above. Although not that • common, a company can expect to find • candidates who meet this requirement.

  28. 12.2 The Normal Model • Using Normal Tables 27 of 45

  29. 12.2 The Normal Model • Example: What is P(-0.5 ≤ Z ≤ 1)? • 0.8413 – 0.3085 = 0.5328

  30. 12.3 Percentiles • Example: • Suppose a packaging system fills boxes such • that the weights are normally distributed with • a µ = 16.3 oz. and σ = 0.2 oz. The package • label states the weight as 16 oz. To what • weight should the mean of the process be • adjusted so that the chance of an • underweight box is only 0.005?

  31. 12.3 Percentiles • Quantile of the Standard Normal • The pth quantile of the standard normal • probability distribution is that value of z such • that P(Z ≤ z ) = p. • Example: Find z such that P(Z ≤ z ) = 0.005. • z = -2.578

  32. 12.3 Percentiles • Quantile of the Standard Normal • Find new mean weight (µ) for process

  33. 4M Example 12.2: VALUE AT RISK • Motivation • Suppose the $1 million portfolio of an • investor is expected to average 10% growth • over the next year with a standard deviation • of 30%. What is the VaR (value at risk) using • the worst 5%?

  34. 4M Example 12.2: VALUE AT RISK • Method • The random variable is percentage change • next year in the portfolio. Model it using the • normal, specifically N(10, 302).

  35. 4M Example 12.2: VALUE AT RISK • Mechanics • From the normal table, we find z = -1.645 for • P(Z ≤ z) = 0.05

  36. 4M Example 12.2: VALUE AT RISK • Mechanics • This works out to a change of -39.3% • µ - 1.645σ = 10 – 1.645(30) = -39.3%

  37. 4M Example 12.2: VALUE AT RISK • Message • The annual value at risk for this portfolio is • $393,000 at 5% (eliminating the worst 5% of • the situations).

  38. 12.4 Departures from Normality • Multimodality. More than one mode suggesting data come from distinct groups. • Skewness. Lack of symmetry. • Outliers. Unusual extreme values.

  39. 12.4 Departures from Normality • Normal Quantile Plot • Diagnostic scatterplot used to determine the appropriateness of a normal model • If data track the diagonal line, the data are normally distributed

  40. 12.4 Departures from Normality • Normal Quantile Plot • Normal Distributions on Both Axes

  41. 12.4 Departures from Normality • Normal Quantile Plot • Distribution on y-axis Not Normal

  42. 12.4 Departures from Normality • Normal Quantile Plot (Diamond Prices) • All points are within dashed curves, normality indicated.

  43. 12.4 Departures from Normality • Normal Quantile Plot (Diamonds of Varying Quality) • Points outside the dashed curves, normality not indicated.

  44. 12.4 Departures from Normality • Skewness • Measures lack of symmetry. K3 = 0 for • normal data.

  45. 12.4 Departures from Normality • Kurtosis • Measures the prevalence of outliers. K4 = 0 • for normal data.

  46. 12.4 Departures from Normality • Prices for Diamonds of Varying Quality

  47. Best Practices • Recognize that models approximate what will happen. • Inspect the histogram and normal quantile plot before using a normal model. • Use z–scores when working with normal distributions.

  48. Best Practices (Continued) • Estimate normal probabilities using a sketch and the Empirical Rule. • Be careful not to confuse the notation for the standard deviation and variance.

  49. Pitfalls • Do not use the normal model without checking the distribution of data. • Do not think that a normal quantile plot can prove that the data are normally distributed. • Do not confuse standardizing with normality.

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