4 A random variable \(X\) has probability density function \(\mathrm { f } ( x ) = \frac { 2 x } { \lambda ^ { 2 } }\) for \(0 < x < \lambda\), where \(\lambda\) is a positive constant.

1. Show that, for any value of \(\lambda , \mathrm { f } ( x )\) is a valid probability density function.
2. Find \(\mu\), the mean value of \(X\), in terms of \(\lambda\) and show that \(\mathrm { P } ( X < \mu )\) does not depend on \(\lambda\).
3. Given that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { \lambda ^ { 2 } } { 2 }\), find \(\sigma ^ { 2 }\), the variance of \(X\), in terms of \(\lambda\). The random variable \(X\) is used to model the depth of the space left by the filling machine at the top of a jar of jam. The model gives the following probabilities for \(X\) (whatever the value of \(\lambda\) ).

   \(0 < X \leqslant \mu - \sigma\)	\(\mu - \sigma < X \leqslant \mu\)	\(\mu < X \leqslant \mu + \sigma\)	\(\mu + \sigma < X < \lambda\)
   0.18573	0.25871	0.36983	0.18573

   A sample of 50 random observations of \(X\), classified in the same way, is summarised by the following frequencies.

   4

   11

   20

   15

4. Carry out a suitable test at the \(5 \%\) level of significance to assess the goodness of fit of \(X\) to these data. Explain briefly how your conclusion may be affected by the choice of significance level.