Druyd:Knowledge

Distribution Characterization

Storyboard

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.

>Model

ID:(310, 0)

Distribution Characterization

Storyboard

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.

Variables

Symbol

Text

Variable

Value

Units

Calculate

MKS Value

MKS Units

Calculations

Equation

Number

First, select the equation:

, then, select the variable:

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable

Given

Calculate

Target :

Equation

To be used

Equations

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

u =sum( P(u_i) * u_i , i ,1, M )

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

f = sum( P(u_i) * f(u_i) , i , 0 , M )

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

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

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

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

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

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

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

u =int( P(u) * u , u ,0, infty )

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

fu =int( P(u) * f(u) , u ,0,infty)

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

@SUM( P(u_i) , i ,1, M ) = 1

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

int( P(u) , u ,0,infty) = 1

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

mDu ^2=@INT( P(u) * ( u - mu )^2 , u )

Examples

Average value of Variables, discrete case

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:

equation

Probability normalization, discrete case

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:

equation

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

Average value of Variables, continuous case

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

equation=3362

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:

equation

Probability normalization, continuous case

As in the discrete case

equation=11434

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

equation

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

Average value of functions, discrete case

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

equation

Average value of functions, continuous case

The ratio of mean values for variables in the discrete case

equation=3363

can be generalized for variable functions

equation

Average Value of a Functions multiplied by Constant

The linearity of the mean values means that the average of a constant for a function

equation=11433

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

equation

Average value of Sum of Functions

The linearity of the mean values means that the average sum of functions of the kind

equation=11433

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

equation

Valor medio de la desviaci n est ndar, caso discreto

A measure of how wide the distribution is is provided by the standard deviation calculated by

equation

Mean value of standard deviation, continuous case

In the discrete case the standard deviation is defined as

equation=3366

which in the continuous limit corresponds to

equation

>Model

ID:(310, 0)