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

We derive the exact distribution of the largest eigenvalue for finite dimensions real Wishart matrices and for the Gaussian Orthogonal Ensemble (GOE). We compare the exact distribution with the Tracy-Widom distribution which arises in many fields as the limiting distribution of the largest eigenvalue of large random matrices. In this regard we show that the Tracy-Widom distribution can be approximated by a properly scaled and shifted Gamma distribution, with great accuracy for the values of common interest in statistical applications.