I'm only having trouble with (formalizing) last two requirements: and .
Find and . Are they independent?
Find also , and .
Now the last two:
is obviously 1, but I don't know how to setup the corresponding definite integral(s).
Similarly for which is reasonably enough (I've peeked at the solutions, but still) ... as would , wouldn't it?
Thank you ... indeed.
Further related question:
I'm told:Let .
What is distribution of ?
"If holds and is a "well behaved" function, monotonous for each in (in ?!),
then ."
How do I find what function is and then use the given formula? The solution given is .
I prefer not to use Z. I'll use U instead.
Use the change of variable theorem:
Make the transformation U = X + Y, V = X. The inverse transformation is X = V, Y = U - V. Then |J| = 1.
Get the joint pdf of U and V: . Integrate wrt to v to get the marginal, which is the required pdf. Note that 0 < y < x => 0 < u - v < v => v < u < 2v and so the region of integration is between the lines v = u/2 and v = u where v > 0:
etc.
Yes.
See for example Walpole, Meyers and Meyers: Probability and Statistics for Engineers and Scientists. It is also explained here: Transformations of Variables (scroll down).