The needle on a broken car speedometer is free to swing, and bounces perfectly off the pin...

The needle on a broken car speedometer is free to swing, and bounces perfectly off the pins at either end, so that if you give it a flick it is equally likely to come to rest at any angle between 0 and π.

(a) What is the probability density, ρ(θ)? Hint: ρ(θ) dθ is the probability that needle will come to rest between θ and (θ + dθ). Graph ρ(θ) as a function of θ, from −π/2 to 3π/2. (Of course, part of this interval is excluded, so ρ is zero there.) Make sure that the total Probability is 1.

(b) Compute θ, θ2, and σ, for this distribution.

(c) Compute sinθ, cosθ, and cos2θ.