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[itex]

|\phi\rangle=\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle

[/itex]

where [itex]0\leq\theta\leq\pi[/itex] and [itex]0\leq\phi\leq 2\pi[/itex], and [itex]0\leq\theta\leq\pi[/itex]. To pick one such state uniformly at random it suffices to draw [itex]\phi[/itex] at random from its domain and [itex]\cos\theta[/itex] uniformly in the range [itex][-1,1] [/itex]. How would you do the equivalent parametrization for an n-dimensional state?