Jan 11, 2016
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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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12.1 Normal Random Variable
Percentage Change in Stock Market Data
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12.1 Normal Random Variable
Prices for One-Carat Diamonds
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12.1 Normal Random Variable
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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12.1 Normal Random Variable
Definition
A continuous random variable whose probability distribution defines a standard bell-shaped curve.
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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.
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12.1 Normal Random Variable
Central Limit Theorem Illustrated
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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)
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12.1 Normal Random Variable
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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12.1 Normal Random Variable
Normal Distributions with Different µ’s
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12.1 Normal Random Variable
Normal Distributions with Different σ’s
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12.1 Normal Random Variable
Standard Normal Distribution (µ = 0; σ2 = 1)
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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)
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12.1 Normal Random Variable
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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12.2 The Normal Model
Normal Model for Stock Market Changes
Set µ = 0.972% and σ = 4.49%.
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12.2 The Normal Model
Normal Model for Diamond Prices
Set µ = $4,066 and σ = $738.
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12.2 The Normal Model
Standardizing to Find Normal ProbabilitiesStart by converting x into a z-score
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xz
12.2 The Normal Model
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
12.2 The Normal Model
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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4M Example 12.1: SATS AND NORMALITY
Method – Use the Normal Model
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4M Example 12.1: SATS AND NORMALITY
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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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.
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12.2 The Normal Model
Using Normal Tables
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12.2 The Normal Model
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
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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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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).
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4M Example 12.2: VALUE AT RISK
Mechanics
From the normal table, we find z = -1.645 for P(Z ≤ z) = 0.05
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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%
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4M Example 12.2: VALUE AT RISK
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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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
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12.4 Departures from Normality
Normal Quantile Plot (Diamond Prices)
All points are within dashed curves, normality indicated.
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12.4 Departures from Normality
Normal Quantile Plot
Points outside the dashed curves, normality not indicated.
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12.4 Departures from Normality
Skewness
Measures lack of symmetry. K3 = 0 for normal data.
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n
zzzK n
332
31
3
...
12.4 Departures from Normality
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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