Finding k in a probability density function

[mnphys]

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.

 

[StoneTemplePython]

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).

Are you sure that you've read and written the question correctly? One possible way to read the above is that Z has a range of (0, 2) and both X and Y have a range of (0,1)...

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?

If you have my above interpretation, then yes you can do this.

 

[mnphys]

The book does say "0 < x" and "y < 1" but maybe you're right and that should be assumed?

It does work out perfectly if you use 0 to 1 as the ranges.

 

[LCKurtz]

I agree. It is common to interpret $0 < x,~y < 1$ to mean both $0 < x < 1$ and $0<y<1$.

 

[mnphys]

Ok. I'll roll with that... Thanks, guys (or possibly gals)!