Copyright © 2011 Pearson Education, Inc. The Normal Probability Model Chapter 12.

Copyright © 2011 Pearson Education, Inc.

The Normal Probability Model

Chapter 12

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)


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Percentage Change in Stock Market Data


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Prices for One-Carat Diamonds


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With the exception of Black Monday, the histogram of market changes is bell-shaped

The histogram of diamond prices is also bell-shaped

Both involve a continuous range of values


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Definition

A continuous random variable whose probability distribution defines a standard bell-shaped curve.


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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.


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Central Limit Theorem Illustrated


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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)


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The Normal Probability Distribution

Defined by the parameters µ and σ2

The mean µ locates the center

The variance σ2 controls the spread


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Normal Distributions with Different µ’s


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Normal Distributions with Different σ’s


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Standard Normal Distribution (µ = 0; σ2 = 1)


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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)


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Probabilities are Areas Under the Curve


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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.


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Normal Model for Stock Market Changes

Set µ = 0.972% and σ = 4.49%.


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Normal Model for Diamond Prices

Set µ = $4,066 and σ = $738.


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Standardizing to Find Normal ProbabilitiesStart by converting x into a z-score


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xz


Standardizing Example: Diamond PricesNormal with µ = $4,066 and σ = $738

Want P(X > $5,000)


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27.1738

066,4000,5000,5000,5$ ZP

XPXP


The Empirical Rule, Revisited


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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?


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Method – Use the Normal Model


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MechanicsA math SAT score of 600 is equivalent to

z = 1. Using the empirical rule, we find that 15.85% of test takers score 600 or better.


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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.


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Using Normal Tables


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Example: What is P(-0.5 ≤ Z ≤ 1)?

0.8413 – 0.3085 = 0.5328


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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?


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12.3 PercentilesQuantile 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


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12.3 PercentilesQuantile of the Standard Normal

Find new mean weight (µ) for process


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52.165758.22.0165758.22.0

16

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%?


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Method

The random variable is percentage change next year in the portfolio. Model it using the normal, specifically N(10, 302).


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Mechanics

From the normal table, we find z = -1.645 for P(Z ≤ z) = 0.05


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Mechanics

This works out to a change of -39.3%

µ - 1.645σ = 10 – 1.645(30) = -39.3%


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Message

The annual value at risk for this portfolio is $393,000 at 5%.


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12.4 Departures from Normality

Multimodality. More than one mode suggesting data come from distinct groups.

Skewness. Lack of symmetry.

Outliers. Unusual extreme values.


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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


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Normal Quantile Plot (Diamond Prices)

All points are within dashed curves, normality indicated.


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Normal Quantile Plot

Points outside the dashed curves, normality not indicated.


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Skewness

Measures lack of symmetry. K3 = 0 for normal data.


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n

zzzK n

332

31

3

...


Kurtosis

Measures the prevalence of outliers. K4 = 0 for normal data.


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3... 44

241

4

n

zzzK n

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.


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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.


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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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Copyright © 2011 Pearson Education, Inc. The Normal Probability Model Chapter 12.

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