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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 an 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. |
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Probability density functions are
indicated for two random variables. Their means are indicted on the
x-axes. |
Let's formalize this. If X is discrete, we define
its expectation as
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[1] |
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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Salsburg (2001)
is a wonderful history of probability and statistics. Degroot and
Schervish (2002)
is a standard university text. Evans, Hastings and Peacock (2000)
is a handy reference with detailed information of numerous
probability distributions.
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