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During the 1990's,
value-at-risk was novel, and computational techniques were widely
discussed in the financial literature. During this period, there was a
popular misperception that a sum of normal random variables is itself
normal. This is often true, but not always. We shall consider a
counterexample shortly.
Let X be an n-dimensional random vector with mean
vector
and covariance matrix
 .
Suppose the marginal distribution of each component
is normal. Let Y be a random variable defined as a
linear
polynomial:
of X, where b is a constant n-dimensional
row vector and a is a scalar. The random variable Y need not
be normal, but there is a condition we can impose on X that
will make it be. That condition is joint-normality.
The joint-normal
distribution (also called the multinormal
or multivariate normal
distribution) can be specified in various ways. Perhaps the simplest is
this: A random vector has a joint-normal distribution if every non-trivial
liner polynomial of the random vector is itself normal. We denote the n-dimensional
joint-normal distribution with mean vector
and covariance matrix
as
 .
If
is positive definite, the probability density function (PDF) is:
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[2] |
where
is the determinant of the covariance matrix.
Exhibit 1 depicts the PDF of a 2-dimensional joint-normal random vector
X with independent components
and
 :
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PDF of 2-dimensional joint-normal random
vector with independent components. |
If we define Y =
+
,
then Y is normal. This is an example of a more general concept from
probability theory called
stability.
If X ~
,
b is a constant
matrix and a is an m-dimensional constant vector,
then:
bX + a ~
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[3] |
This generalizes the analogous 1-dimensional
property of univariate
normal distributions.
Now let’s consider how a random vector might fail to be
joint-normal despite each of its components being marginally normal. Let
X be a 2-dimensional random vector with components
and
.
Let
and Z be independent standard
normal random variables, and define
 ,
where the sign function returns 1 if
and returns
 if
 .
In this case, both
and
are standard normal, but the vector X, of which they are
components, is not joint-normal. Its PDF is graphed in Exhibit 2:
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PDF of a 2-dimensional random vector. Both
of its components have marginal distributions that are normal, but
the random vector is not joint-normal. |
In this case, the random variable Y = X1 +
X2 is not normal.
Instead, it has the PDF illustrated in Exhibit 3.
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The PDF of the sum of two components of a
random vector. Both components are marginally normal, but the random
vector is not joint-normal. Its PDF is shown in Exhibit 2. The sum
of the two components is dramatically non-normal. |
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 Brownian
motion A simple continuous stochastic process that is widely used in physics
and finance for modeling random behavior that evolves over time.
central
limit theorem A theorem that explains why the normal
distribution plays such an important role in probability theory.
chi-squared distribution
If you square a normal random variable, the result is a
chi-squared random variable.
Cholesky matrix
A lower-triangular matrix that acts as a matrix "square root" for a positive
definite matrix.
correlation
A parameter, related to covariance, that indicates the tendency for two random
variables to "move together" of "co-vary."
expected value A parameter
describing the "center of gravity" of a distribution.
kurtosis A parameter describing the peakedness and tails of a
distribution.
linear
polynomial of a random vector A random variable or random
vector that is defined as a linear polynomial of a random vector.
lognormal
distribution A random variable is
lognormal if its logarithm is normal.
multicollinear
A covariance matrix is muticollinear if it is "almost" singular.
normal
distribution Perhaps the most
important probability distribution for probability and statistics.
positive definite
matrix A real symmetric matrix, all of whose eigenvalues are real and positive.
principal
component analysis A technique for orthogonalizing a random
vector.
quantile A notion from
probability that can be used as a parameter.
skewness A parameter that
describes the lack of symmetry of a distribution.
stable
Paretian distribution A non-normal stable distribution.
standard deviation A
parameter describing the dispersion of a distribution.
uniform
distribution A continuous
probability distribution that has constant probability on a finite
interval. |
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Ads by Contingency Analysis
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Evans, Hastings and Peacock (2000)
is a handy reference with detailed information on numerous probability distributions. Johnson and Wichern (2002)
is a multivariate statistics text with a full chapter on the joint-normal distribution.
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 Risk
Correlations 06 May 1999
Measuring the risk of rare events and properties of the
joint-normal distribution. |
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