Methods of finding the distribution of g(x,y).

[Ohad]

Hi all,

I was wondering what methods each of you like to use when he/she tries to find the distribution of a function of random variables?

For example:

v~N(0,1) and x~{1 w.p 0.5 , -1 w.p 0.5}
y=x*v
y=x+v
y=x^2+v
etc

From my experience it seems that there is no one way which is the best way to attack all of these problems.
Using the "wrong" method can lead you to a mathematical adventure while using the "right" method will get you there in two lines.

So how would you attack these problems?

Thanks,
Ohad.

 

[Alejandro Canete]

http://en.wikipedia.org/wiki/Multivariate_kernel_density_estimation

 

[Ohad]

http://en.wikipedia.org/wiki/Multivariate_kernel_density_estimation

Hi thanks, but I was talking about analytic methods (obviously applicable only for certain simple cases).

 

[Alejandro Canete]

Hi thanks, but I was talking about analytic methods (obviously applicable only for certain simple cases).

Ohh, sorry, cant help you with that

 

[Ohad]

Ohh, sorry, cant help you with that

No problem, it actually very common and everyone here have done it hundreds of times.

For example in the example I gave y=x*v than one way to do it is to simple stop...and logically understand that this is a Gaussian mixture since x can be only two values multiplied by the distribution of v.

It can also be achieved by f(y)=f(y|x=1)*f(x=1)+f(y|x=-1)*f(x=-1) , now f(y|x)=f(v) hence once again we get a Gaussian mixture.

In the case of y=x+v one can use the convolution of the two distributions but many times it will be much more messy than using the conditional distribution or some other method.

Hope this made helped me get my point across better.

 

[Tsotne]

Hi all,

I was wondering what methods each of you like to use when he/she tries to find the distribution of a function of random variables?

For example:

v~N(0,1) and x~{1 w.p 0.5 , -1 w.p 0.5}
y=x*v
y=x+v
y=x^2+v
etc

From my experience it seems that there is no one way which is the best way to attack all of these problems.
Using the "wrong" method can lead you to a mathematical adventure while using the "right" method will get you there in two lines.

So how would you attack these problems?

Thanks,
Ohad.

Are you talking about the joint probability distribution functions or copulas??? I didn't quite well understood the question. Or for the marginal distribution functions. For both cases finding the distribution is the hypothesis testing whether the distribution against which the data has been checked is rejected or accepted. You can apply Chi-Square test as well as Kolmogorov test. The last one is much more powerful though. In this case however, I think (if I got the question correctly) you are interested in joint distribution functions of random variables. The same tests I mention above holds.

http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm
http://en.wikipedia.org/wiki/Kolmogorov–Smirnov_test

You can apply the same test checking for multiple dimensional distribution hypothesis with little modification. With 2-3 dimensions it is especially fairly easily imaginable.