7. Some children are asked to mark the centre of a scale 10 cm long. The position they choose is indicated by the variable \(X\), where \(0 \leq X \leq 10\). Initially, \(X\) is modelled as a random variable with a continuous uniform distribution.

1. Find the mean and the standard deviation of \(X\). It is suggested that a better model would be the distribution with probability density function $$f ( x ) = c x , 0 \leq x \leq 5 , \quad f ( x ) = c ( 10 - x ) , 5 < x \leq 10 , \quad f ( x ) = 0 \text { otherwise. }$$
2. Write down the mean of \(X\).
3. Find \(c\), and hence find the standard deviation of \(X\) in this model.
4. Find \(\mathrm { P } ( 4 < X < 6 )\). It is then proposed that an even better model for \(X\) would be a Normal distribution with the mean and standard deviation found in parts (b) and (c).
5. Use these results to find \(\mathrm { P } ( 4 < X < 6 )\) in the third model.
6. Compare your answer with (d). Which model do you think is most appropriate? (1 mark)