- Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
(6) - A continuous random variable \(X\) has the probability density function \(\mathrm { f } ( x )\) shown in Figure 1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{58e5aa9e-f177-48ad-8bb8-54c0e2c21e6d-07_591_689_358_630}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
- Show that \(\mathrm { f } ( x ) = 4 - 8 x\) for \(0 \leqslant x \leqslant 0.5\) and specify \(\mathrm { f } ( x )\) for all real values of \(x\).
- Find the cumulative distribution function \(\mathrm { F } ( x )\).
- Find the median of \(X\).
- Write down the mode of \(X\).
- State, with a reason, the skewness of \(X\).