Standard +0.3 This is a straightforward variance calculation from a given pdf. Students must find k using the normalization condition, then compute E(X) and E(X²) using standard integration. The polynomial integrand (6x - x²) is simple, and the question explicitly tells them what answer to show, removing any uncertainty about whether their work is correct. This is slightly easier than average as it's a routine application of formulas with no conceptual challenges.
6 The probability density function, f, of a random variable \(X\) is given by
$$f ( x ) = \begin{cases} k \left( 6 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$
where \(k\) is a constant.
State the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 9 } { 5 }\).
6 The probability density function, f, of a random variable $X$ is given by
$$f ( x ) = \begin{cases} k \left( 6 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant.\\
State the value of $\mathrm { E } ( X )$ and show that $\operatorname { Var } ( X ) = \frac { 9 } { 5 }$.\\
\hfill \mbox{\textit{CAIE S2 2021 Q6 [6]}}