- #1
mnphys
- 10
- 0
Homework Statement
Let X, Y, and Z have the joint probability density function f(x,y,z) = kxy2z for 0 < x, y < 1, and 0 < z < 2 (it is defined to be 0 elsewhere). Find k.
Homework Equations
Not sure how to type this in bbcode but: Integrate f(x,y,z) = kxy2z over the ranges of x (zero to infinity) , y (negative infinity to 1), and z (zero to two) and set k so that the result is equal to 1 (by the definition of a PDF).
The Attempt at a Solution
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The problem with this is that I keep running into divergent integrals, and I'm not sure how to avoid this. I have done all my work on paper and trying to type it all out into Word's equation editor is driving me insane but I can take photos of the work I've done if you want proof that I have DEFINITELY tried to work this out... like for pages and pages.
For example, if I start by integrating f(x,y,z) = xy2z respective to x, I end up with a non-divergent improper integral (the second term is equal to 0, but the first term takes the limit of x2y2z/2 as x approaches infinity). Let's move on and substitute "t" for x but remember that "t" is approaching infinity. If I then integrate respective to y, it gets worse - substituting "u" for y as y approaches negative infinity, that looks like t2z/6 - t2u3z/6 (remembering that t is approaching infinity and u is approaching negative infinity). Now integrating with respect to z leaves me with a really ugly equation that involves multiple terms all approaching infinity... yuck.
So... how do I resolve these divergent integrals? The book's answer is 3, and if I just set y and x's limits of integration to 1 (upper bound) and 0 (lower bound) then I get that answer, but... I can't do that, can I? Or I could change the order of integration, but no matter what order I try, I end up with at least one term involving taking the limit of a positive exponential as it approaches infinity.
Please, don't think I'm asking you to solve this for me - I really am not. Any hint as to how I can avoid these divergent integrals, or what I'm doing wrong, is all I'm asking for. If anyone would like me to upload photos of my work on this, I can do that.