(a) Find the standard deviation of the distribution in Example 1.2.

(b) What is the probability that a photograph, selected at random, would show a distance x more than one standard deviation away from the average?

Question

(a) Find the standard deviation of the distribution in Example 1.2.

(b) What is the probability that a photograph, selected at random, would show a distance x more than one standard deviation away from the average?

Accepted Answer

(a)

[latex] \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} [/latex].

[latex] 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 [/latex] ⇒ [latex] σ = \frac{2h}{3\sqrt{5} } = 0.2981 [/latex] h .

(b)

[latex] 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) [/latex] .

[latex] x_+ ≡ \left\langle x\right\rangle + σ = 0.3333 h + 0.2981 h = 0.6315 h [/latex] .

[latex] x_- ≡ \left\langle x\right\rangle - σ = = 0.3333 h + 0.2981 h = 0.0352h [/latex] .

[latex] P = 1 - \sqrt{0.6315} + \sqrt{0.0352} = 0.393 [/latex].

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

This Question has been Answered.

Already have an account?

Login

Related Answered Questions