Moderate -0.8 This is a straightforward S2 question testing basic properties of continuous uniform distributions. Part (a)(i) requires simple integration to find k=1/(a+b), part (a)(ii) is a standard proof by integration of the mean formula, and part (b) applies these results to a numerical example with routine probability calculations. All steps are textbook exercises with no problem-solving or novel insight required, making it easier than average.
The continuous random variable \(T\) follows a rectangular distribution with probability density function given by
$$\mathrm { f } ( t ) = \left\{ \begin{array} { l c }
k & - a \leqslant t \leqslant b \\
0 & \text { otherwise }
\end{array} \right.$$
Express \(k\) in terms of \(a\) and \(b\).
Prove, using integration, that \(\mathrm { E } ( T ) = \frac { 1 } { 2 } ( b - a )\).
The error, in minutes, made by a commuter when estimating the journey time by train into London may be modelled by the random variable \(T\) with probability density function
$$\mathrm { f } ( t ) = \left\{ \begin{array} { c c }
\frac { 1 } { 10 } & - 4 \leqslant t \leqslant 6 \\
0 & \text { otherwise }
\end{array} \right.$$
Write down the value of \(\mathrm { E } ( T )\).
Calculate \(\mathrm { P } ( T < - 3\) or \(T > 3 )\).
3
\begin{enumerate}[label=(\alph*)]
\item The continuous random variable $T$ follows a rectangular distribution with probability density function given by
$$\mathrm { f } ( t ) = \left\{ \begin{array} { l c }
k & - a \leqslant t \leqslant b \\
0 & \text { otherwise }
\end{array} \right.$$
\begin{enumerate}[label=(\roman*)]
\item Express $k$ in terms of $a$ and $b$.
\item Prove, using integration, that $\mathrm { E } ( T ) = \frac { 1 } { 2 } ( b - a )$.
\end{enumerate}\item The error, in minutes, made by a commuter when estimating the journey time by train into London may be modelled by the random variable $T$ with probability density function
$$\mathrm { f } ( t ) = \left\{ \begin{array} { c c }
\frac { 1 } { 10 } & - 4 \leqslant t \leqslant 6 \\
0 & \text { otherwise }
\end{array} \right.$$
\begin{enumerate}[label=(\roman*)]
\item Write down the value of $\mathrm { E } ( T )$.
\item Calculate $\mathrm { P } ( T < - 3$ or $T > 3 )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2008 Q3 [8]}}