We consider the same device as the previous problem, but this time we are interested in th...

We consider the same device as the previous problem, but this time we are interested in the x-coordinate of the needle point—that is, the “shadow,” or “projection,” of the needle on the horizontal line.

(a) What is the probability density ρ(x)? Graph ρ(x) as a function of x, from −2r to + 2r, where r is the length of the needle. Make sure the total probability is 1. Hint: ρ(x) dx is the probability that the projection lies between x and (x + dx). You know (from Problem 1) the probability that θ is in a given range; the question is, what interval dx corresponds to the interval dθ?

(b) Compute x, x2, and /, for this distribution. Explain how you could have obtained these results from part (c) of Problem 1.

Problem 1

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.