Let D be the unit disc of

Let D be the unit disc of R^2, that is, D = {(x, y) : x^2 + y^2 < 1}. Then the random point (X, Y ) is uniformly distributed on D, if its distribution is given by the density function:
f (x, y)=1/π if (x, y) ∈ D
=0 otherwise

Show that the polar coordinates R ∈ [0, 1] and  tetha∈ [0, 2π] of the point are independent.

Let's use the change of variables x = r cos θ, y = r sin θ.

So, the Jacobian ∂(x, y)/∂(r, θ) = r.
==> f_R,θ (r, θ) = (1/π) * r = r/π for r in [0, 1], θ in [0, 2π], and 0 otherwise.

Now, it's easy to check independence.

First of all, we compute the marginal densities:
f_R (r) = ∫(θ = 0 to 2π) (r/π) dθ = 2r for r in [0, 1] (and 0 otherwise).
f_θ (θ) = ∫(r = 0 to 1) (r/π) dr = 1/(2π) for θ in [0, 2π] (and 0 otherwise).

Since f_R,θ (r, θ) = f_R (r) * f_θ (θ), we conclude that R and θ are independent.

I hope this helps!