Standard +0.8 This is a Further Pure 2 transformation of variables question requiring the Jacobian method (dy/dx = 3x², so dx/dy = 1/(3x²) = 1/(3y^(2/3))), substitution into the original pdf, and then integration of y^(2/3) to find E(Y). While the technique is standard for FP2, it requires careful algebraic manipulation and is more conceptually demanding than typical A-level statistics questions.
8 The continuous random variable \(X\) has probability density function f given by
$$f ( x ) = \begin{cases} \frac { 1 } { 6 } x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Show that \(Y\) has probability density function g given by
$$g ( y ) = \begin{cases} \frac { 1 } { 18 } y ^ { - \frac { 1 } { 3 } } & 8 \leqslant y \leqslant 64 \\ 0 & \text { otherwise } \end{cases}$$
Find \(\mathrm { E } ( Y )\).
8 The continuous random variable $X$ has probability density function f given by
$$f ( x ) = \begin{cases} \frac { 1 } { 6 } x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
The random variable $Y$ is defined by $Y = X ^ { 3 }$. Show that $Y$ has probability density function g given by
$$g ( y ) = \begin{cases} \frac { 1 } { 18 } y ^ { - \frac { 1 } { 3 } } & 8 \leqslant y \leqslant 64 \\ 0 & \text { otherwise } \end{cases}$$
Find $\mathrm { E } ( Y )$.
\hfill \mbox{\textit{CAIE FP2 2013 Q8 [9]}}