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Let X be a random variable. We denote the
expected value (expectation
or mean) of X as either
 or E(X).
It represents the (probability-weighted) average value for the random
variable. You might think of the mean of a distribution as being analogous
to a center of mass.
This is illustrated in Exhibit 1. Probability density
functions (PDFs) are indicated for two random variables. The means of each are
indicated on the x-axes. In engineering, a center of mass is a
"balancing point." By visual inspection, you can confirm that the
indicated means in Exhibit 1 appear to be "balancing points" for the
two PDFs. Let's formalize this. If X is discrete, we define
its expectation as
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[1] |
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Probability density functions are
indicated for two random variables. Their means are indicted on the
x-axes. |
where
 is the
probability function (PF) of X. If X is continuous, we
replace the summation with an integral and define
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[2] |
where
is now the
probability density function (PDF) of X.
The concept of
expected value generalizes to multiple dimensions. Consider a random
vector X:
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[3] |
Its
mean vector, denoted
 or E(X),
is simply the vector of the expected values of its components:
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[4] |
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 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.
normal distribution
A continuous probability distribution whose probability density
function has a "bell" shape.
quantile A notion from
probability that can be used as a parameter.
skewness A parameter that
describes the lack of symmetry of a 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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http://www.riskglossary.com
copyright © Glyn A. Holton,
2004
Although the information in
this website has been presented with care and obtained from sources the author believes to be reliable, there is no guarantee that it is
accurate. Such information may be incomplete, condensed, outdated or presented with
errors. The content of the website is for information purposes only. It is
provided gratuitously, so the author shall not be liable under any
theory for any damages suffered by any user. The author does not
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investment advice. |
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