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Quantum entanglement plays a crucial role in quantum information, quantumteleportation and quantum computation. The information about the entanglementcontent between subsystems of the composite system is encoded in the Schmidteigenvalues. We derive here closed expressions for the spectral density ofSchmidt eigenvalues for all three invariant classes of random matrix ensembles.We also obtain exact results for average von Neumann entropy. We find thatmaximum average entanglement is achieved if the system belongs to thesymplectic invariant class.

Entanglement in random pure states: Spectral density and average von Neumann entropy