Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 122

Other location parameters are, for continuous distributions, the median, that
value m for which F X (m) = , and the quantiles, those values x q for which F X (x q )=

q, and, for discrete distributions, the modes, those values M for which p X (M)

½

≥

p X (x) for any real x.
Similar to the location parameters, another kind of useful information about the
distribution is given by the dispersion parameters, from which the most frequently
used is the variance.
Variance of the random variable X is the nonnegative number

2

VX ðÞ ¼ EX ð EX Þ :

So the variance of X is the expected value of a measure of deviation of X from
its location parameter EX.
From this de nition follow that

EX 2

2

VX ðÞ ¼ ðÞ EX :

nition is the
square function. To bring the measurement to the same scale of X, instead of the
variance, is used to measure dispersion the standard deviation, a parameter

The symmetric measure of deviation from EX employed in this de

˃ (X)

de ned as the square root of the variance. This means

1 = 2

2

r ðÞ ¼ X EX ð EX Þ :

A.4 Properties of the Expected Value and the Variance

The main property of the concept of expected value is linearity:

EX ð þ Y Þ ¼ EX ðÞþ EY ðÞ

and

EcX ðÞ ¼ cE X ðÞ for any real c :

For any event A of S and 1 A the random variable de ned by

1A s ðÞ ¼ 1ifs 2 A and 1 A s ðÞ ¼ 0ifs 2 S n A ;

E1 ðÞ ¼ pA ðÞ:

By this correspondence, the concept of expected value may replace the concept
of probability of an event.