91Ó°ÊÓ

The proportion of the total number of conceivable outcomes to the number of options in an exhaustive collection of equally likely outcomes that cause a given occurrence.

Because \({\rm{X}}\)is uniformly distributed on \({\rm{(0,360)}}\), its pdf \({\rm{(x)}}\) can be written as follows for all \({\rm{x}}\) in this interval:

\({\rm{f(x) = }}\frac{{\rm{1}}}{{{\rm{360 - 0}}}}{\rm{ = }}\frac{{\rm{1}}}{{{\rm{360}}}}\)

Overall, \({\rm{f(x)}}\) can be written as: \({\rm{f(x) = }}\left\{ {\begin{array}{*{20}{c}}{\frac{{\rm{1}}}{{{\rm{360}}}}{\rm{0\poundsx\pounds360}}}\\{{\rm{\;0 otherwise\;}}}\end{array}} \right.\)

\({\rm{X}}\)'s expected value can be expressed as: \(\begin{array}{*{20}{c}}{{\rm{E(X) = \`o }}_{{\rm{ - \currency}}}^{\rm{\currency}}{\rm{nx \times f(x) \times dx}}}\\{{\rm{ = \`o }}_{\rm{0}}^{{\rm{360}}}{\rm{nx \times }}\frac{{\rm{1}}}{{{\rm{360}}}}{\rm{ \times dx}}}\\{{\rm{ = }}\frac{{\rm{1}}}{{{\rm{360}}}}{\rm{\`o }}_{\rm{0}}^{{\rm{360}}}{\rm{nx \times dx}}}\\{{\rm{ = }}\frac{{\rm{1}}}{{{\rm{360}}}}\left( {\frac{{{{\rm{x}}^{\rm{2}}}}}{{\rm{2}}}} \right)_{\rm{0}}^{{\rm{360}}}}\\{{\rm{ = }}\frac{{\rm{1}}}{{{\rm{720}}}}\left( {{\rm{36}}{{\rm{0}}^{\rm{2}}}{\rm{ - }}{{\rm{0}}^{\rm{2}}}} \right)}\\{{\rm{E(X) = 180}}}\end{array}\)

Definition: A continuous \({\rm{rv's}}\) expected or mean value. \({\rm{f(x)}}\) is \({\rm{X}}\)with pdf

\({\rm{E}}\left( {{{\rm{X}}^{}}} \right)\)\({\rm{ = }}\mathop {\rm{\`o }}\nolimits_{{\rm{ - \currency}}}^{\rm{\currency}} {{\rm{x}}^{\rm{2}}}{\rm{ \times f(x) \times dx}}\)

Now consider the following argument:

The standard deviation \({{\rm{\sigma }}_{\rm{x}}}\) and variance \({\rm{V(X)}}\)of a \({\rm{rv's}}\) \({\rm{X}}\)with a specified pdf can be stated as: \(\begin{array}{*{20}{c}}{{\rm{V(X) = E}}\left( {{{\rm{X}}^{\rm{2}}}} \right){\rm{ - (E(X)}}{{\rm{)}}^{\rm{2}}}}\\{{{\rm{\sigma }}_{\rm{x}}}{\rm{ = }}\sqrt {{\rm{V(X)}}} }\end{array}\)

We begin by calculating \({\rm{E}}\left( {{{\rm{X}}^{\rm{2}}}} \right)\).

\(\begin{array}{*{20}{c}}{{\rm{E}}\left( {{{\rm{X}}^{\rm{2}}}} \right)}&{{\rm{ = }}\mathop {\rm{\`o }}\nolimits_{{\rm{ - \currency}}}^{\rm{\currency}} {{\rm{x}}^{\rm{2}}}{\rm{ \times f(x) \times dx}}}\\{}&{{\rm{ = }}\mathop {\rm{\`o }}\nolimits_{\rm{0}}^{{\rm{360}}} {{\rm{x}}^{\rm{2}}}{\rm{ \times }}\frac{{\rm{1}}}{{{\rm{360}}}}{\rm{ \times dx}}}\\{}&{{\rm{ = }}\frac{{\rm{1}}}{{{\rm{360}}}}\mathop {\rm{\`o }}\nolimits_{\rm{0}}^{{\rm{360}}} {{\rm{x}}^{\rm{2}}}{\rm{ \times dx}}}\\{}&{{\rm{ = }}\frac{{\rm{1}}}{{{\rm{360}}}}\left( {\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right)_{\rm{0}}^{{\rm{360}}}}\\{}&{{\rm{ = }}\frac{{\rm{1}}}{{{\rm{1080}}}}\left( {{\rm{36}}{{\rm{0}}^{\rm{3}}}{\rm{ - }}{{\rm{0}}^{\rm{2}}}} \right)}\\{{\rm{E}}\left( {{{\rm{X}}^{\rm{2}}}} \right)}&{{\rm{ = 43200}}}\end{array}\)

Using the above proposition, we can now write:

\(\begin{array}{*{20}{c}}{{\sigma _x}}&{ = \sqrt {V(X)} }\\{}&{ = \sqrt {E\left( {{X^2}} \right) - {{(E(X))}^2}} }\\{}&{\sqrt {43200 - {{(180)}^2}} }\end{array}\)

\(\sqrt {43200 - 32400} \)

\(\sqrt {10800} \)

\({\sigma _x} = 103.9230\)

We can apply the following proposition because \({\rm{Y( = h(X)}}\) is a linear function of \({\rm{X}}\):

The expected value and standard deviation of \({\rm{h(X)}}\) meet the following conditions for \({\rm{h(X) = aX + b}}\).

\(\begin{array}{*{20}{c}}{{\rm{E(h(X)) = aE(X) + b}}}\\{{{\rm{\sigma }}_{{\rm{h(x)}}}}{\rm{ = a}}{{\rm{\sigma }}_{\rm{x}}}}\end{array}\)

We are informed that:

\({\rm{Y = }}\frac{{{\rm{2\pi }}}}{{{\rm{360}}}}{\rm{ \times X - \pi }}\)

As a result, the predicted value of \({\rm{Y}}\) is:

\(\begin{array}{*{20}{c}}{{\rm{E(Y)}}}&{{\rm{ = }}\frac{{{\rm{2\pi }}}}{{{\rm{360}}}}{\rm{ \times E(X) - \pi }}}\\{}&{{\rm{ = }}\frac{{{\rm{2\pi }}}}{{{\rm{360}}}}{\rm{ \times 180 - \pi }}}\\{}&{{\rm{ = }}\frac{{{\rm{360}}}}{{{\rm{360}}}}{\rm{ \times \pi - \pi }}}\\{{\rm{E(Y)}}}&{{\rm{ = 0}}}\end{array}\)

The standard deviation of \({\rm{Y}}\) is calculated as follows:

\(\begin{array}{*{20}{c}}{{{\rm{\sigma }}_{\rm{y}}}{\rm{ = }}\frac{{{\rm{2\pi }}}}{{{\rm{360}}}}{\rm{ \times }}{{\rm{\sigma }}_{\rm{x}}}}\\{{\rm{ = }}\frac{{{\rm{2\pi }}}}{{{\rm{360}}}}{\rm{ \times (103}}{\rm{.923)}}}\\{{{\rm{\sigma }}_{\rm{y}}}{\rm{ = (0}}{\rm{.57735)\pi }}}\end{array}\)