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Distribution Characterization
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There are a number of parameters that can be calculated with a probability distribution such as mean values and standard deviation for both discrete and continuous distributions.

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Average value of Variables, discrete case
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If the values are given

u_1, u_2, \ldots, u_M

with its corresponding probabilities

P(u_1), P(u_2), \ldots, P(u_M)

With this, an average value can be calculated:

$ \bar{u} =\displaystyle\sum_{ i =1}^ M P(u_i) u_i $

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Probability normalization, discrete case
Equation

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In that case you can define discrete values

u_1, u_2, \ldots, u_M

with its corresponding probabilities

P(u_1), P(u_2), \ldots, P(u_M)

the latter must be standardized:

$ \displaystyle\sum_{ i =1}^ M P(u_i) = 1$

which means that all possible outcomes are included in the probability function P(u).

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Average value of Variables, continuous case
Equation

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The average that is calculated as the sum of the discrete u_i values weighted with the probability P(u_i)

$ \bar{u} =\displaystyle\sum_{ i =1}^ M P(u_i) u_i $



it has its corresponding expression for the continuous case. In that case you can define value u with its corresponding probability P(u). With this, an average value can be calculated:

$ \bar{u} =\displaystyle\int du\,P(u)\,u $

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Probability normalization, continuous case
Equation

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As in the discrete case

$ \displaystyle\sum_{ i =1}^ M P(u_i) = 1$



u values can be defined with their corresponding probabilities P(u), the latter must be normalized:

$ \displaystyle\int P(u) du = 1$

which means that all possible outcomes are included in the probability function P(u).

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Average value of functions, discrete case
Equation

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The relationship of mean values for variables can be generalized for functions of variables

$ \overline{f} =\displaystyle\sum_{ i =1}^ M P(u_i) f(u_i) $

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Average value of functions, continuous case
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The ratio of mean values for variables in the discrete case

$ \overline{f} =\displaystyle\sum_{ i =1}^ M P(u_i) f(u_i) $



can be generalized for variable functions

$ \overline{f} =\displaystyle\int P(u) f(u) du$

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Average Value of a Functions multiplied by Constant
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The linearity of the mean values means that the average of a constant for a function

$ \overline{f} =\displaystyle\int P(u) f(u) du$



is equal to the product of the constant for the mean value of the function:

$\overline{cf}=c\overline{f}$

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Average value of Sum of Functions
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The linearity of the mean values means that the average sum of functions of the kind

$ \overline{f} =\displaystyle\int P(u) f(u) du$



is equal to the mean value of each of the functions:

$\overline{f+g}=\overline{f}+\overline{g}$

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Valor medio de la desviación estándar, caso discreto
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A measure of how wide the distribution is is provided by the standard deviation calculated by

$\overline{(\Delta u)^2}=\displaystyle\sum_{i=1}^M P(u_i)(u_i-\bar{u})^2$

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Mean value of standard deviation, continuous case
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In the discrete case the standard deviation is defined as

$\overline{(\Delta u)^2}=\displaystyle\sum_{i=1}^M P(u_i)(u_i-\bar{u})^2$



which in the continuous limit corresponds to

$ \overline{(\Delta u)^2} =\displaystyle\int P(u) ( u - \bar{u} )^2 du$

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