February/March 2026 Paper 62

The widths, $X$ cm, of the stones found on a certain beach are modelled by the probability density function given by

where $k$ and $a$ are constants. Show that $k=\frac{6}{a^3}$.

Without calculation, give an expression for $\mathrm{E}(X)$ in terms of $a$.

Find $\mathrm{P}\left(X<\frac{1}{4}a\right)$, showing that this probability is independent of $a$.

Use your answer to part (c)(i) to find $\mathrm{P}\left(\frac{1}{4}a<X<\frac{3}{4}a\right)$.