Sampling a hemisphere using an arbitary distribtuion

I am writing a ray tracer and I wish to fire rays from a point **p** into a hemisphere above that point according to some distribution.

1) I have derived a method to uniformly sample within a solid angle (defined by theta) above **p** [1]

\(\phi = 2\pi\xi_1\)

\(\alpha = \arccos (1-(1-\cos(\theta))\xi_2)\)

\(x = \sin(\alpha)\cos\phi\)

\(y = \sin(\alpha)\sin\phi\)

\(z = -\cos(\alpha)\)

Where\(\xi\) is a uniform random number

That works and Im pretty happy with that. But my question is what happens if I do not want a uniform distribution.

I have used the algorithm on page 27 from [2] and I can draw samples from a piecewise arbitrary distribution. However if I simply say:

\(\alpha = \arccos (1-(1-cos(\theta)) \beta_1)\)

Where $\beta$ is a random number generated from an arbiatry distribution.
It doesn't behave nicely...What am I doing wrong? Thanks in advance. I really really need help on this