If θ is a continuous random variable which is uniformly distributed between 0 and π, write down an expression for p(θ). IIence find the value of the following averages: (i) ⟨θ⟩ (ii) ⟨θ−π/2)
(iii) ⟨θ^2)
(iv) ⟨θ ^n) (for the case n≥0 )
(v) ⟨cosθ⟩
(vi) ⟨sinθ⟩ (vii) ⟨∣cosθ∣⟩
(viii) ⟨cos^2θ⟩
(ix) ⟨sin^2θ⟩
(x) ⟨cos^2 θ+sin^2 θ⟩ Check that your answers are what you expect. b) If particle velocitios are distributod isotropically, how are their angles distributod? Is the angle between the velocity vector and a fixed axis (chosen by you) distributed uniformly? Why? Answer these questions for the case of a 2- and 3-dimensional world.