Catalan numbers are the moments of the semicircle law

ProofAdvanced⏱ 1 hourRandom Matricessemicirclemomentscatalanwigner

Problem

Let $\rho_{sc}(x) = \frac{1}{2\pi}\sqrt{4-x^2}$ on $[-2,2]$ .

Verify $\int_{-2}^2\rho_{sc} = 1$ .
Show the odd moments vanish and that the $2k$ -th moment is the Catalan number $C_k = \frac{1}{k+1}\binom{2k}{k}$ .
Verify $C_1 = 1$ , $C_2 = 2$ , $C_3 = 5$ and interpret the second moment.

Write a complete argument. Every inequality you use should be one you could state precisely.

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