1 Design engineers are simulating the load on a particular part of a complex structure. They intend that the simulated load, measured in a convenient unit, should be given by the random variable \(X\) having probability density function $$f ( x ) = 12 x ^ { 3 } - 24 x ^ { 2 } + 12 x , \quad 0 \leqslant x \leqslant 1 .$$

1. Find the mean and the mode of \(X\).
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\). $$\text { Verify that } \mathrm { F } \left( \frac { 1 } { 4 } \right) = \frac { 67 } { 256 } , \mathrm {~F} \left( \frac { 1 } { 2 } \right) = \frac { 11 } { 16 } \text { and } \mathrm { F } \left( \frac { 3 } { 4 } \right) = \frac { 243 } { 256 } .$$ The engineers suspect that the process for generating simulated loads might not be working as intended. To investigate this, they generate a random sample of 512 loads. These are recorded in a frequency distribution as follows.

   Load \(x\)	\(0 \leqslant x \leqslant \frac { 1 } { 4 }\)	\(\frac { 1 } { 4 } < x \leqslant \frac { 1 } { 2 }\)	\(\frac { 1 } { 2 } < x \leqslant \frac { 3 } { 4 }\)	\(\frac { 3 } { 4 } < x \leqslant 1\)
   Frequency	126	209	131	46

3. Use a suitable statistical procedure to assess the goodness of fit of \(X\) to these data. Discuss your conclusions briefly.