Standard +0.8 This question requires finding E(T) where T = L² + ½S with L continuous and S discrete. Students must find the constant w using integration, calculate E(L²) using the transformation formula for continuous variables, calculate E(S) for the discrete variable, then apply linearity of expectation and independence. The multi-step nature, mixing continuous and discrete distributions, and requiring integration of l³ makes this moderately challenging for Further AS level.
5 Participants in a school jumping competition gain a total score for each jump based on the length, \(L\) metres, jumped beyond a fixed point and a mark, \(S\), for style.
\(L\) may be regarded as a continuous random variable with probability density function
$$\mathrm { f } ( l ) = \left\{ \begin{array} { c c }
w l & 0 \leq l \leq 15 \\
0 & \text { otherwise }
\end{array} \right.$$
where \(w\) is a constant.
\(S\) may be regarded as a discrete random variable with probability function
$$\mathrm { P } ( S = s ) = \left\{ \begin{array} { c l }
\frac { 1 } { 15 } s & s = 1,2,3,4,5 \\
0 & \text { otherwise }
\end{array} \right.$$
Assume that \(L\) and \(S\) are independent.
The total score for a participant in this competition, \(T\), is given by \(T = L ^ { 2 } + \frac { 1 } { 2 } S\)
Show that the expected total score for a participant is \(114 \frac { 1 } { 3 }\)
5 Participants in a school jumping competition gain a total score for each jump based on the length, $L$ metres, jumped beyond a fixed point and a mark, $S$, for style.\\
$L$ may be regarded as a continuous random variable with probability density function
$$\mathrm { f } ( l ) = \left\{ \begin{array} { c c }
w l & 0 \leq l \leq 15 \\
0 & \text { otherwise }
\end{array} \right.$$
where $w$ is a constant.\\
$S$ may be regarded as a discrete random variable with probability function
$$\mathrm { P } ( S = s ) = \left\{ \begin{array} { c l }
\frac { 1 } { 15 } s & s = 1,2,3,4,5 \\
0 & \text { otherwise }
\end{array} \right.$$
Assume that $L$ and $S$ are independent.
The total score for a participant in this competition, $T$, is given by $T = L ^ { 2 } + \frac { 1 } { 2 } S$
Show that the expected total score for a participant is $114 \frac { 1 } { 3 }$
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics Q5 [5]}}