Is there an issue with the correlation function?

Question

I am trying to find the correlation between Uniform R.V.s using the Correlation command in mathematica, but it seems that either I am implementing it wrong or the software cannot find the answer. My code is pretty straight forward:

Correlation[UniformDistribution[], Sqrt[1-(UniformDistribution[])^2]]

I also tried

Correlation[x, Sqrt[1-(x)^2], x \[Distributed] UniformDistribution[]]

I am aware that I can implement directly the definition of correlation and compute it in terms of expectation, but I was wondering if it was possible directly through this function.

Answer 1

Update: A much more compact approach...

dist = TransformedDistribution[{x, Sqrt[1 - x^2]}, x \[Distributed] UniformDistribution[]];

Correlation[dist, 1, 2]
(* (8 - 3 π)/Sqrt[32 - 3 π^2] *)

End of update

I think you want the correlation between $X$ and $\sqrt{1-X^2}$ where $X \sim Uniform(0,1)$.

μX = Mean[UniformDistribution[]]
(* 1/2 *)

varX = Variance[UniformDistribution[]]
(* 1/12 *)

distY = TransformedDistribution[Sqrt[1 - x^2], x \[Distributed] UniformDistribution[]];
μY = Mean[distY]
(* π/4 *)

varY = Variance[distY]
(* 1/48 (32-3 π^2) *)

covXY = Integrate[x Sqrt[1 - x^2], {x, 0, 1}] - μX μY
(* 1/48 (32-3 π^2) *)

ρ = covXY/Sqrt[varX varY] // FullSimplify
(* (8-3 π)/Sqrt[32-3 π^2] *)
ρ // N

As a check consider random samples from a uniform distribution:

SeedRandom[12345];
n = 100000;
z = RandomVariate[UniformDistribution[], n];
Correlation[z, Sqrt[1 - z^2]]
(* -0.9214182413747128` *)

Answer 2

In general you can get correlation matrix for multivariate symbolic distribution like this:

corr = Correlation[ ProductDistribution[UniformDistribution[], TransformedDistribution[Sqrt[1 - x^2], 
   x \[Distributed] UniformDistribution[]]]]

{{1, 0}, {0, 1}}

I believe you are looking for corr[[2,1]].