5
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{54bf68ab-7934-432a-890f-20093082ab07-12_446_1105_242_479}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) and Figure 1 shows a sketch of \(\mathrm { f } ( x )\) where
$$f ( x ) = \left\{ \begin{array} { c c }
k ( 1 - \cos x ) & 0 \leqslant x \leqslant 2 \pi
0 & \text { otherwise }
\end{array} \right.$$
- Show that \(k = \frac { 1 } { 2 \pi }\)
The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(\mathrm { E } ( Y ) = \mathrm { E } ( X )\)
The probability density function of \(Y\) is \(g ( y )\), where
$$g ( y ) = \frac { 1 } { \sigma \sqrt { 2 \pi } } e ^ { - \frac { 1 } { 2 } \left( \frac { y - \mu } { \sigma } \right) ^ { 2 } } \quad - \infty < y < \infty$$
Given that \(\mathrm { g } ( \mu ) = \mathrm { f } ( \mu )\) - find the exact value of \(\sigma\)
- Calculate the error in using \(\mathrm { P } \left( \frac { \pi } { 2 } < Y < \frac { 3 \pi } { 2 } \right)\) as an approximation to \(\mathrm { P } \left( \frac { \pi } { 2 } < X < \frac { 3 \pi } { 2 } \right)\)