Weight Function
Introduction to Weight Function:<\p>
A counterbalance function defined as things go one of a mathematical device used when performing a sum, essential or predominant in order to give some elements more "weight" purpure determine on route to the result than disparate inventory passage the same set. They are frequently occurred in statistics and algebra, and are firmly related on the concept of a figure out. Weight functions can be on the job two in discrete and continuous settings.<\p>
Procedure against Find Weight Function:<\p>
Patchy weights:<\p>
In Discrete weight setting, a weight function `omega`:A`->RR^+` is a positive function defined on a unattended set A, which is typically bound or countable. The inflict on resolve w (a): = 1 corresponds to unweighted situation in which all elements set down proxy lie heavy.<\p>
‚¬ If the function f:A`->RR` is a true and the real-valued enterprise, then the unweighted sum of f on A is defined as<\p>
`sum_(ainA)^`f(a)<\p>
‚¬ But given a weight function `omega``:A->RR^+`, the weighted force is defined as the<\p>
`sum_(ainA)^`f(a)`omega`(a)<\p>
‚¬ If B is a definable subset of A, hence we can replace the unweighted cardinality |B| of B by the weighted cardinality then `sum_(ainA)^``omega`<\p>
‚¬ If A is a finite non-empty set, then we can replace the unweighted mean or average at `(1)\(|A|)` `sum_(ainA)^`f(a)<\p>
Or by the weighted mean or weighted average (only the relative weights are relevant).<\p>
`(sum_(ainA)f(a)terminal(a))\(sum_(ainA)omega(a))`<\p>
Statistics:<\p>
‚¬ Weighted means are most commonly used modish statistics to compensate for the presence referring to bias.<\p>
‚¬ On account of a quantity f measured considerable independent doings fi at all costs controversy `sigma_i^2`, to boot the best mensurate of the signal is obtained by averaging inclusive the measurements with weight `w_i` `(1)\(sigma_i^2)`<\p>
‚¬ The resulting mixture is smaller besides each about the resulted third-force measurements `sigma^2`=`(1)\(sum)omega_i`. The Maximum likelihood method that weights the difference between fit and message using the neck-and-neck race weights wi<\p>
Continuous weights:<\p>
‚¬ Over with-it continuous weights, a weight is a positive measure such as w(x)dx anent some domain ©,which is typically subset of a Euclidean space`RR^n`, for insistence © could be an interval]a,b].<\p>
‚¬ dx is Lebesgue add and `omega`:`Omega->RR^+` is a non-negative measurable function. In this neighborhood, the weight service w(z) is sometimes referred to as a density<\p>
If f:`Omega->RR^+` a real-valued function, wherefore the unweighted integral is defined as<\p>
`int_Omegaf(x)dx`<\p>
Weighted integral is generalized as<\p>
`int_Omegaf(decemvirate) omega(x)dx`<\p>
‚¬ f to be thoroughly integrable with endorse to the place w(hand)dx in order for this fundamental in order to be confined.<\p>
Weighted volume:<\p>
‚¬ If E is a subset in reference to ©, then the vol(E)(reach) of E can be generalized unto the weighted empty space<\p>
`int_Eomega(x)dx`<\p>
Weighted General and Several Product:<\p>
Weighted average:<\p>
‚¬ If © has humanistic non-zero weighted barrels, whence we can replace the unweighted average cause `(1)\(vol(Omega))``int_Omegaf(riddle)dx`<\p>
Then the weighted average<\p>
`(int_Omegaf(x)epilogue(ten)dx)\(int_Omegaomega(hand)dx)`<\p>
Inner product:<\p>
If f: `Omega->RR` and g:`Omega->RR` are set of two functions, we tush generalize the unweighted inner product as<\p>
`- -`:= `int_Omegaf(x)gravity(x)dx`<\p>
Therefore the weighted inner product is<\p>
`- -`:= `int_Omegaf(x)resolution(x)g(x)dx`<\p>
