Determining what method to use to derive probability density functions of random variables

Question

I have just learned how to derive pdfs of random variables. The method of transformations is much easier for me. However it is confusing to me when it is okay to use this method instead of the method of distribution functions. Correct anything I say from this point on:

To use the method of transformations the function must be monotonic, meaning it is constantly increasing or decreasing. To do this you would take the first derivative to see if this holds.

So for example, X and Y are independent, continuous random variables, both uniformly distributed on the interval [0, 2]. If we had to compute the probability density function of U=X+Y we can't use the method of transformations, correct? This is since the derivative of X+Y is 1 and is not constantly increasing or decreasing.

How about X having an exponential distribution with mean $\frac{1}{2}$. The pdf of Y we are computing which is equal to $\sqrt{X}$. We could use the method of transformations here since the derivative of $\sqrt{X}=\frac{1}{2\sqrt{x}}$ which is always decreasing.

All of this seems logical to me but I want to see if my thinking is correct...

Answer 1

To use the method of transformations the function must be monotonic, meaning it is constantly increasing or decreasing. To do this you would take the first derivative to see if this holds.

This sounds about right. Note that you can cheat a bit with the method of transformations if a function isn't too non-monotonic, but that's not super important for this discussion and your statement is a good opening bid.

So for example, X and Y are independent, continuous random variables, both uniformly distributed on the interval [0, 2]. If we had to compute the probability density function of U=X+Y we can't use the method of transformations, correct? This is since the derivative of X+Y is 1 and is not constantly increasing or decreasing.

Let's back up for a second -- what's relevant is not whether the derivative is increasing or decreasing, but rather whether the derivative is positive or negative. Also, you're talking about a derivative of $f(x,y) = x+ y$ -- a derivative with respect to which variable, though? You can always discuss partial derivatives as a proxy, but that's not what's in the letter of the law of the method of transformations.

Also, the method of transformations is best suited to dealing with functions of one variable, such as $e^X$ or $\sqrt X$ as you suggest later, not two variables like $X + Y$ or $XY$. (The multivariate issue is the real sticking point for this example.)

How about X having an exponential distribution with mean $\frac{1}{2}$. The pdf of Y we are computing which is equal to $\sqrt{X}$. We could use the method of transformations here since the derivative of $\sqrt{X}=\frac{1}{2\sqrt{x}}$ which is always decreasing.

Similar to the above: it's not relevant that it's always decreasing; it's relevant that it's always positive. If the derivative of the function were, say, $\sin(x) + 2$, that would also be fine; it's always positive, though it certainly is not always decreasing.