Translated uniform unit ball pdf

Question

Let $X$ be a random variable distributed uniformly in the unit ball, then we know that the pdf w.r.t. spherical coordinates is a constant multiplied by the Jacobian inside the sphere (and $0$ outside). Then is it true that $Y = X + (0,\rho,0), \rho \ne 0$ is distributed according to a pdf that is not constant anymore?

Answer 1

I figured it out, I had a mistake in my calculations. As expected the translation shouldn't change the pdf, the original being: $$p(x,y,z) = \Theta(\rho-\sqrt{x^2+y^2+z^2})\frac{3}{4\pi\rho^3}$$ and the translated: $$q(x,y-\rho,z)= \Theta(\rho-\sqrt{x^2+(y-\rho)^2+z^2})\frac{3}{4\pi\rho^3}$$ Where $\Theta$ is the Heaviside step function. The constant remains the same as expected.