7
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e90d172-0292-4bb6-9d4a-506669076d4e-3_385_385_982_246}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e90d172-0292-4bb6-9d4a-506669076d4e-3_385_380_982_669}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e90d172-0292-4bb6-9d4a-506669076d4e-3_390_378_977_1087}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e90d172-0292-4bb6-9d4a-506669076d4e-3_390_391_977_1503}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e90d172-0292-4bb6-9d4a-506669076d4e-3_392_391_1475_370}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e90d172-0292-4bb6-9d4a-506669076d4e-3_392_387_1475_872}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e90d172-0292-4bb6-9d4a-506669076d4e-3_392_389_1475_1375}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{figure}
Each of the random variables \(T , U , V , W , X , Y\) and \(Z\) takes values between 0 and 1 only. Their probability density functions are shown in Figs 1 to 7 respectively.
- (a) Which of these variables has the largest median?
(b) Which of these variables has the largest standard deviation? Explain your answer. - Use Fig. 2 to find \(\mathrm { P } ( U < 0.5 )\).
- The probability density function of \(X\) is given by
$$f ( x ) = \begin{cases} a x ^ { n } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$
where \(a\) and \(n\) are positive constants.
(a) Show that \(a = n + 1\).
(b) Given that \(\mathrm { E } ( X ) = \frac { 5 } { 6 }\), find \(a\) and \(n\).