AQA
Further AS Paper 2 Statistics
Specimen
Q5
5 marks
Standard +0.8
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 \(114 \frac { 1 } { 3 }\)