If x1,x2, .. Xn be random variable with density function of fxi(xi) and y=w1*x1+w2*x2+...+wn*xn which wi are constant value.
What is fy(y)?
Can we calculate it directly? If not how about an estimation?
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if your xi are _independent_ random variables, see
- convolution product
http://en.wikipedia.org/wiki/Convolution
- moment generating function
http://en.wikipedia.org/wiki/Moment-generating_function
if they are not independent variables, exact computation of the multi-dimensional integral may still be possible in very particular cases, otherwise ... numerical integration will be required
the alternative is to build the cumulative distribution function of y just by drawing enough independent samples from the joint distribution of the (xi) ; won't give you a closed formula but if you just need an approximation of the density, this will be an easy way (if you have access to the joint density of the vector (xi) !)
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hello Fabrica
my xi are not independent .
if we have just x1 and x2 so y=w1*x1+w2*x2 how can We calculate probability density function of y ?
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Just a little elementary calculus (of course, this usually does not lead to a closed form solution) or am i missing something in our question ?
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Hello Fabrice
Thanks again for answering . I got that it does not lead to a closed form solution. but i need the derivative of fy(y) to w2 . How can i calculate d (fy(y))/dw2 ?
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there must be something i'm missing here
this sounds like elementary calculus : the density fy(y) is just the derivative in y of the cumulative distribution, and you calculate the derivative in w2 of this density ...
all you need to know is
http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign
and ... well, the resulting expression may be quite ugly !
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