Compare uniform with other distributions

7 Two continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}
T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases} \end{array}$$

1. Sketch on the same axes the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). [You should not use graph paper or attempt to plot points exactly.]
2. Explain in everyday terms the difference between the two random variables.
3. Find the value of \(t\) such that \(\mathrm { P } ( T > t ) = 0.2\).

5 The continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & \mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } & - 2 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}
T : & \mathrm { g } ( x ) = \begin{cases} \frac { 5 } { 64 } x ^ { 4 } & - 2 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases} \end{array}$$

1. Sketch, on the same axes, the graphs of f and g .
2. Describe in everyday terms the difference between the distributions of the random variables \(S\) and \(T\). (Answers that comment only on the shapes of the graphs will receive no credit.)
3. Calculate the variance of \(T\).

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)