Standard +0.8 This is a multi-part question involving a piecewise continuous probability distribution requiring integration across different domains. Part (a) requires finding the upper quartile by solving F(x)=0.75 across two regions, part (b) involves careful probability calculation across the boundary, and part (c) requires computing E(X²) and E(X) separately then solving for their ratio. While the individual integrations are standard A-level techniques, the piecewise nature and multi-step reasoning across three parts elevates this above average difficulty.
5. The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \begin{cases} \frac { 4 } { 3 x ^ { 3 } } & 1 \leq x < 2 \\ \frac { 1 } { 12 } x & 2 \leq x \leq 4 \\ 0 & \text { otherwise } \end{cases}$$
a) Find the upper quartile of \(X\).
b) Find \(\mathrm { P } \left( \frac { 1 } { 2 } < X \leq 3 \right)\)
c) Find the value of \(a\) for which \(\mathrm { E } \left( X ^ { 2 } \right) = a \mathrm { E } ( X )\).
5. The continuous random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} \frac { 4 } { 3 x ^ { 3 } } & 1 \leq x < 2 \\ \frac { 1 } { 12 } x & 2 \leq x \leq 4 \\ 0 & \text { otherwise } \end{cases}$$
a) Find the upper quartile of $X$.\\
b) Find $\mathrm { P } \left( \frac { 1 } { 2 } < X \leq 3 \right)$\\
c) Find the value of $a$ for which $\mathrm { E } \left( X ^ { 2 } \right) = a \mathrm { E } ( X )$.\\
\hfill \mbox{\textit{SPS SPS FM Statistics 2020 Q5 [11]}}