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We derive efficient recursive formulas giving the exact distribution of thelargest eigenvalue for finite dimensional real Wishart matrices and for theGaussian Orthogonal Ensemble (GOE). In comparing the exact distribution withthe limiting distribution of large random matrices, we also found that theTracy-Widom law can be approximated by a properly scaled and shifted Gammadistribution, with great accuracy for the values of common interest instatistical applications.

Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution