Question 1.2: (a) Find the standard deviation of the distribution in Examp...

(a)

$\left\langle x^{2}\right\rangle=\int_{0}^{h} x^{2} \frac{1}{2 \sqrt{h x}} d x=\left.\frac{1}{2 \sqrt{h}}\left(\frac{2}{5} x^{5 / 2}\right)\right|_{0} ^{h}=\frac{h^{2}}{5}$.

$σ^2 = \left\langle x^{2}\right\rangle – \left\langle x\right\rangle^2 = \frac{h^{2}}{5} – \Bigl(\frac{h}{3}\Bigr) ^2 = \frac{4}{45}h^2$ ⇒ $σ = \frac{2h}{3\sqrt{5} } = 0.2981$ h .

(b)

$P=1-\int_{x_{-}}^{x_{+}} \frac{1}{2 \sqrt{h x}} d x=1-\left.\frac{1}{2 \sqrt{h}}(2 \sqrt{x})\right|_{x_{-}} ^{x_{+}}=1-\frac{1}{\sqrt{h}}\left(\sqrt{x_{+}}- \sqrt{x_{-}}\right)$ .

$x_+ ≡ \left\langle x\right\rangle + σ = 0.3333 h + 0.2981 h = 0.6315 h$ .

$x_- ≡ \left\langle x\right\rangle – σ = = 0.3333 h + 0.2981 h = 0.0352h$ .

$P = 1 – \sqrt{0.6315} + \sqrt{0.0352} = 0.393$.