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Chapter 12
The
Normal
Probability
Model
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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
Prices for One-Carat Diamonds
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12.1 Normal Random Variable
Percentage Change in Stock Market
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12.1 Normal Random Variable
X-ray Measurements of Bone Density
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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
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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
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
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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12.1 Normal Random Variable
Normal Distributions with Different µ’s
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12.1 Normal Random Variable
Normal Distributions with Different σ2’s
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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 Diamond Prices
Set µ = $4,066 and σ = $738.
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12.2 The Normal Model
Normal Model for Stock Market Changes
Set µ = 0.94% and σ = 4.32%.
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12.2 The Normal Model
Normal Model for Bone Density Scores
Set µ = -1.53 and σ = 1.3.
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12.2 The Normal Model
Standardizing to Find Normal Probabilities
Start by converting x into a z-score
z
x

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12.2 The Normal Model
Standardizing Example: Diamond Prices
Normal with µ = $4,066 and σ = $738
Want P(X > $5,000)
5,000  4,066
 X   5,000   


P X  $5,000  P

 1.27 
  P Z 

738
 



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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 or more?
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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
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.
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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 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
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12.3 Percentiles
Quantile of the Standard Normal
Find new mean weight (µ) for process
16  
 2.5758    16  0.22.5758  16.52
0.2
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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% (eliminating the worst 5% of
the situations).
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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
Normal Distributions on Both Axes
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12.4 Departures from Normality
Normal Quantile Plot
Distribution on y-axis Not Normal
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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 (Diamonds of Varying Quality)
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.
z  z  ...z
K3 
n
3
1
3
2
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n
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12.4 Departures from Normality
Kurtosis
Measures the prevalence of outliers. K4 = 0
for normal data.
z  z  ...  z
K4 
3
n
4
1
4
2
4
n
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12.4 Departures from Normality
Prices for Diamonds of Varying Quality
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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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