The Marchenko–Pastur density integrates to one

ProofAdvanced⏱ 1 hourRandom Matricesmarchenko-pasturintegralsubstitution

Problem

For $\lambda\in(0,1]$ and $\sigma=1$ , verify that

\int_a^b \frac{\sqrt{(b-x)(x-a)}}{2\pi\lambda x}\,dx = 1, \qquad a = (1-\sqrt\lambda)^2,\ b = (1+\sqrt\lambda)^2 .

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

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