91Ó°ÊÓ

$$\begin{gathered} 1=\underset{-∞}{\overset{∞}{∫}}\mathrm{ÒÏ}\left(\mathrm{θ}\right)d\mathrm{θ}\backslash \mathrm{hfill} \\ 1=\underset{0}{\overset{\mathrm{Ï€}}{∫}}\mathrm{ÒÏ}\left(\mathrm{θ}\right)d\mathrm{θ}\backslash \mathrm{hfill} \end{gathered}$$

It is clear from this integration that the integrand must be constant,

So,

$\mathrm{ÒÏ}\left(\mathrm{θ}\right)=\frac{1}{\mathrm{Ï€}},0â\copyright ½\mathrm{θ}â\copyright ½\mathrm{Ï€}$

$\mathrm{ÒÏ}\left(\mathrm{θ}\right)=0$,otherwise

The localid="1658292402403" $\mathrm{ÒÏ}\left(\mathrm{θ}\right)$ for the graph is zero except in the interval localid="1658292406704" $0â\copyright ½\mathrm{θ}â\copyright ½\mathrm{Ï€}$

Calculating the expectation values.

$$\begin{gathered} ⟨\mathrm{θ}âŸ\copyright =\underset{-∞}{\overset{∞}{∫}}\mathrm{θ}\mathrm{ÒÏ}\left(\mathrm{θ}\right)d\mathrm{θ} \\ ⟨\mathrm{θ}âŸ\copyright =\underset{0}{\overset{\mathrm{Ï€}}{∫}}\mathrm{θ}\frac{1}{\mathrm{Ï€}}d\mathrm{θ} \\ ⟨\mathrm{θ}âŸ\copyright =\frac{1}{\mathrm{Ï€}}{\left.\frac{{\mathrm{θ}}^2}{2}\right|}_0^{\mathrm{Ï€}} \end{gathered}$$

$⟨\mathrm{θ}âŸ\copyright =\frac{\mathrm{Ï€}}{2}$

Expectation value is given by,

$$\begin{gathered} ⟨{\mathrm{θ}}^2âŸ\copyright =\underset{-∞}{\overset{∞}{∫}}{\mathrm{θ}}^2\mathrm{ÒÏ}\left(\mathrm{θ}\right)d\mathrm{θ} \\ ⟨{\mathrm{θ}}^2âŸ\copyright =\underset{0}{\overset{\mathrm{Ï€}}{∫}}{\mathrm{θ}}^2\frac{1}{\mathrm{Ï€}}d\mathrm{θ} \\ ⟨{\mathrm{θ}}^2âŸ\copyright =\frac{1}{\mathrm{Ï€}}{\left.\frac{{\mathrm{θ}}^3}{3}\right|}_0^{\mathrm{Ï€}} \\ ⟨{\mathrm{θ}}^2âŸ\copyright =\frac{{\mathrm{Ï€}}^2}{3} \end{gathered}$$

Standard deviation is given by,

$$\begin{gathered} \mathrm{σ}=\sqrt{⟨{\mathrm{θ}}^2âŸ\copyright -{⟨\mathrm{θ}âŸ\copyright }^2} \\ \mathrm{σ}=\sqrt{\frac{{\mathrm{Ï€}}^2}{3}-\frac{{\mathrm{Ï€}}^2}{4}} \\ \mathrm{σ}=\frac{\mathrm{Ï€}}{2\sqrt{3}} \end{gathered}$$

For $\sin \mathrm{θ}$ ,

$$\begin{gathered} ⟨\sin \mathrm{θ}âŸ\copyright =\underset{-∞}{\overset{∞}{∫}}\sin \mathrm{θ}.\mathrm{ÒÏ}\left(\mathrm{θ}\right)d\mathrm{θ} \\ ⟨\sin \mathrm{θ}âŸ\copyright =\frac{1}{\mathrm{Ï€}}\underset{0}{\overset{\mathrm{Ï€}}{∫}}\sin \mathrm{θ}d\mathrm{θ} \\ ⟨\sin \mathrm{θ}âŸ\copyright =-{\left.\frac{1}{\mathrm{Ï€}}\cos \mathrm{θ}\right|}_0^{\mathrm{Ï€}} \\ ⟨\sin \mathrm{θ}âŸ\copyright =\frac{2}{\mathrm{Ï€}} \end{gathered}$$

For $\cos \mathrm{θ}$,

$⟨\cos \mathrm{θ}âŸ\copyright =\underset{-∞}{\overset{∞}{∫}}\cos \mathrm{θ}\mathrm{ÒÏ}\left(\mathrm{θ}\right)d\mathrm{θ}$

$$\begin{gathered} ⟨\cos \mathrm{θ}âŸ\copyright =\frac{1}{\mathrm{Ï€}}\underset{0}{\overset{\mathrm{Ï€}}{∫}}\cos \mathrm{θ}d\mathrm{θ} \\ ⟨\cos \mathrm{θ}âŸ\copyright ={\left.\frac{1}{\mathrm{Ï€}}\sin \mathrm{θ}\right|}_0^{\mathrm{Ï€}} \\ ⟨\cos \mathrm{θ}âŸ\copyright =0 \end{gathered}$$

And for ${\cos }^2\mathrm{θ}$

$$\begin{gathered} ⟨\cos \mathrm{θ}âŸ\copyright =\frac{1}{\mathrm{Ï€}}\underset{0}{\overset{\mathrm{Ï€}}{∫}}\cos \mathrm{θ}d\mathrm{θ} \\ ⟨\cos \mathrm{θ}âŸ\copyright ={\left.\frac{1}{\mathrm{Ï€}}\sin \mathrm{θ}\right|}_0^{\mathrm{Ï€}} \\ ⟨\cos \mathrm{θ}âŸ\copyright =0 \end{gathered}$$

Since, ${\cos }^2\mathrm{θ}=\frac{1}{2}\cos \left(2\mathrm{θ}\right)+\frac{1}{2}$

Therefore,

$⟨{\cos }^2\mathrm{θ}âŸ\copyright =\frac{1}{\mathrm{Ï€}}\underset{0}{\overset{\mathrm{Ï€}}{∫}}\left(\frac{1}{2}\cos \left(2\mathrm{θ}\right)+\frac{1}{2}\right)d\mathrm{θ}$

For the first integral, taking

$2\mathrm{θ}=u$

Then,$d\mathrm{θ}=\frac{1}{2}du$$d\mathrm{θ}=\frac{1}{2}du$

And the bound of integration becomes$\mathrm{θ}→\mathrm{π},u→2\mathrm{π}$and as$\mathrm{θ}→0,u→0$

$$\begin{gathered} ⟨{\cos }^2\mathrm{θ}âŸ\copyright =\frac{1}{2\mathrm{Ï€}}\underset{0}{\overset{2\mathrm{Ï€}}{∫}}\cos udu+{\left.\frac{1}{2\mathrm{Ï€}}\right|}_0^{2\mathrm{Ï€}} \\ ⟨{\cos }^2\mathrm{θ}âŸ\copyright ={\left.\frac{1}{2\mathrm{Ï€}}\sin u\right|}_0^{2\mathrm{Ï€}}+\frac{1}{2} \\ ⟨{\cos }^2\mathrm{θ}âŸ\copyright =0+\frac{1}{2} \\ ⟨{\cos }^2\mathrm{θ}âŸ\copyright =\frac{1}{2} \end{gathered}$$

Hence the solution is :

$$\begin{gathered} \mathrm{ÒÏ}\left(\mathrm{θ}\right)=\left\{\begin{array}{ll}\frac{1}{\mathrm{Ï€}},& 0≤\mathrm{θ}≤\mathrm{Ï€}\\ 0& otherwise\end{array}\right. \\ ⟨\mathrm{θ}âŸ\copyright =\frac{\mathrm{Ï€}}{2},⟨{\mathrm{θ}}^2âŸ\copyright =\frac{{\mathrm{Ï€}}^2}{3},\mathrm{σ}=\frac{\mathrm{Ï€}}{2\sqrt{3}} \\ ⟨\sin \mathrm{θ}âŸ\copyright =\frac{2}{\mathrm{Ï€}},⟨\cos \mathrm{θ}âŸ\copyright =0,⟨{\cos }^2\mathrm{θ}âŸ\copyright =\frac{1}{2}. \end{gathered}$$