Marginal Densities of 3 Joint Normal r.v.s in Polar Coordinates

[tritan.ex47]

Hey all,

So fun question. X, Y and Z are independent ~ N(0,sigma^2).

x = r*sin(phi)*cos(theta)
y = r*sin(phi)*sin(theta)
z = r*cos(phi)
0 <= phi <= pi
0 <= theta <= 2*pi
dx dy dz = (r^2)*sin(phi) dr dtheta dphi

And the question asks us to find the joint pdf, and the marginal densities of phi, theta and r.

So, finding the join pdf is easy enough. Due to convolution, fxyz(x,y,z) = fx(x)*fy(y)*fz(z), so the transformation we have:

fxyz(x,y,z) = fx(r*sin(phi)*cos(theta))*fy(r*sin(phi)*sin(theta))*fz(r*cos(phi))*(r^2)*sin(phi), where fx(), fy() and fz() are the pdfs of N(0, sigma^2), and (r^2)*sin(phi) is the jacobian.

However, coming up with the marginal densities has been somewhat problematic. I tried starting with both r and theta, but I'm creating some really nasty integrals. Even putting just e^((r*sin(phi)*cos(theta))^2) into wolframalpha produces a "no results found..." (e.g. integrating with respect to phi), so I'm wondering if I'm missing some identities? Scrolling through the information I can find (such as trig identities on wikipedia), nothing stands out. However, I know other students in my class have gotten somewhere...so either these integrals are doable, or I'm missing some simplifications?

Any thoughts, hints or help? Thanks everyone!

[BGM]

Can you simplify the joint pdf again to see what you have get?

[tritan.ex47]

Hey there BGM,

Sorry, I'm not sure what you mean. You mean you want me to write out the full joint pdf, or are you just wondering whether things can be simplified further from what I've posted?

If the latter, you can pull out the 1/((sigma^3)*2pi*sqrt(2pi)), and you can combine the exponents in each of the normal pdfs, to get something like

(r^2)*sin(phi) * e^((-((r*sin(phi)*cos(theta))^2)/2sigma^2)-(((r*sin(phi)*sin(theta))^2)/2sigma^2)-(((r*cos(phi))^2)/2sigma^2)

But I'm still having my main problem of just trying to integrate this with respect to either r, phi or theta.

[BGM]

you can still simplify the exponent -

[tritan.ex47]

So you're saying something like the following?

(r^2)*sin(phi) * e^(-(1/(2sigma^2))*(((r*sin(phi)*cos(theta))^2) + ((r*sin(phi)*sin(theta))^2) + ((r*cos(phi))^2)))

to

(r^2)*sin(phi) * e^(-(1/(2sigma^2))*(x^2 + y^2 + z^2)) i.e. the original form

to

(r^2)*sin(phi) * e^(-(1/(2sigma^2))*(r^2))

?

[BGM]

Yes, you can also directly go from above to the bottom by the trigonometric identity .

Now you can factorize the joint pdf into 3 products and you just need to find out the normalizing constants for each marginal by the integration.

[tritan.ex47]

Yep, sounds good BGM. I think I've got it now. Thanks for your help!