Blundell Thermal Physics · Problem 3.5

Probability

Physics — Statistical Mechanics · uniform-distribution-averages

Problem

If θ\theta is a continuous random variable which is uniformly distributed between 0 and π\pi, write down an expression for P(θ)P(\theta). Hence find the value of the following averages: (a) ⟨θ⟩\langle \theta \rangle; (b) ⟨θ−π2⟩\langle \theta - \frac{\pi}{2} \rangle; (c) ⟨θ2⟩\langle \theta^2 \rangle; (d) ⟨θn⟩\langle \theta^n \rangle (for the case n≥0n \geq 0); (e) ⟨cos⁡θ⟩\langle \cos\theta \rangle; (f) ⟨sin⁡θ⟩\langle \sin\theta \rangle; (g) ⟨∣cos⁡θ∣⟩\langle |\cos\theta| \rangle; (h) ⟨cos⁡2θ⟩\langle \cos^2\theta \rangle; (i) ⟨sin⁡2θ⟩\langle \sin^2\theta \rangle; (j) ⟨cos⁡2θ+sin⁡2θ⟩\langle \cos^2\theta + \sin^2\theta \rangle. Check that your answers are what you expect.

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