Consider a random $n \times n$ unitary matrix, i.e. whose probability measure is the Haar measure for $\mathrm{U}(n)$. What is the marginal probability distribution for a single element of this matrix? What about for a random orthogonal matrix?
Since the elements of a unitary matrix must have norm $\leq 1$, the probability density function must be compactly supported on $[-1, 1]$, so it can't be analytic.
This question is similar but (I think) asks for the joint marginal distribution between two different entries, whereas my question is simpler and only asks for the marginal distribution for a single element.