Mixed continuous-discrete problems

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 \(114 \frac { 1 } { 3 }\)

9 The continuous random variable \(X\) has the cumulative distribution function shown below. $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0
\frac { 1 } { 62 } \left( 4 x ^ { 3 } + 6 x ^ { 2 } + 3 x \right) & 0 \leq x \leq 2
1 & x > 2 \end{array} \right.$$ The discrete random variable \(Y\) has the probability distribution shown below. The random variables \(X\) and \(Y\) are independent.
Find the exact value of \(\mathrm { E } \left( X ^ { 3 } + Y \right)\).

6. The continuous random variable \(X\) has the cumulative distribution function shown below. $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0
\frac { 1 } { 62 } \left( 4 x ^ { 3 } + 6 x ^ { 2 } + 3 x \right) & 0 \leq x \leq 2
1 & x > 2 \end{array} \right.$$ The discrete random variable \(Y\) has the probability distribution shown below. The random variables \(X\) and \(Y\) are independent.
Find the exact value of \(\mathrm { E } \left( X ^ { 3 } + Y \right)\).
[0pt] [6 marks]