Moment generating function - ratio with two variables?

Question

I'm not sure if this is possible - but let see:

I want to find the moment generating function (mgf) of x1/(x1+x2) given that I know the mgf of both x1 and x2.

Can this be done using mgfs? - any references??
thanks.

Answer 1

In theory, if X1 and X2 are independent, generating the MGF of a combination of the two random variables from their individual MGFs can almost always be done, because, under normal circumstances, the moment generating function contains all the information you might need about the probability distribution (that's why it can generate all the moments :-). If X1 and X2 are not independent, then the relationship between them is lost and the individual MGFs don't have enough information.

Whether it is easy to do in any specific case, is a separate issue, but take a look at:
http://www.fmi.uni-sofia.bg/vesta/virtual_labs/expect/expect5.html

More generally, it takes a convolution rather than a simple algebraic relationship to compute the MGF of the combine random variable. The fact that your equation requires a division suggests that this is the case here.

Answer 2

searching a is straightforward; all you want to attraction to close is that for any mgf, M_x(0) = a million. in case you already know the prompt generating function, you'll get the function function. in case you already know the function function, you may extract the pdf with an inversion. the first section is straightforward: the function function is phi_x(t) = M_x(it) = a / (3e^{-it} - 2). The inversion formula must be on your textbook; it really is the grotesque one with the vital from -infinity to infinity. it is also on the Wikipedia web page for "function function (probability concept)". (there's a reason we do not coach this stuff in severe college.) have exciting!