I just started this topic and have a question:
For a positive continuous random variable X, write down the PDF of Y = X^2 in terms of the PDF of X.
So; writing the PDF of X, I get
P(0<X<∞) =integral from 0 to ∞ fx(x)dx
Is this correct? For Y=X^2, wouldn't the pdf be the exact same thing, since the question asks for positive continuous random variables anyway?
Is there a general method, approach I should have about this. I find pdfs to be confusing, the textbooks are heavy on jargon
And also, please see the attached image for my other question. I don't understand how to compute the integral, is there some trick?
I do believe that
to find fx(x) I integrate the joint pdf, with respect to x with the bounds set as the range of Y. But this leaves me with a very complex integration
Similarly for fy(y). Is there some trick?
Any help is greatly appreciated.
Re: Writing PDFs
In order to find the PDF for Y = X^2 you need to use something like the transformation theorem of the characteristic equation. Do you have a particular PDF in mind? Here is the transformation theorem (scroll down to change of variables formula):
Transformations of Variables
With regards to your other question, look at the definitions for expectation and marginal densities. The marginal density is used to "integrate out" the other variable so that you get a PDF in terms of one variable.
Also remember that E[XY] = Integral over region xy*f(x,y)dxdy.