# What is the distribution of the angle between two random vectors?

Let $$x,y\stackrel{\text{i.i.d.}}{\sim}\mathcal{N}(0,I_d)$$ be two random $$d$$-dimensional standard normal distributed vectors and let $$\theta$$ denote the angle between them. ($$I_d$$ denotes the $$d$$-dimensional identity matrix).

Do you know about a probability distribution that models either

• the distribution of $$\theta\in[0,\pi]$$, or
• the distribution of $$\cos(\theta)\in[-1,1]$$, or
• the distribution of $$\frac{\cos(\theta)+1}{2}\in[0,1]$$?

A paper (on page 18, second paragraph) says that the third option is distributed according to $$Beta(d/2,d/2)$$. Can you derive that? Or can you point me to a resource proving it? Or might it be that they're just using an approximation?