Find quartiles or percentiles

1 A railway company is investigating operations at a junction where delays often occur. Delays (in minutes) are modelled by the random variable \(T\) with the following cumulative distribution function. $$\mathrm { F } ( t ) = \begin{cases} 0 & t \leqslant 0
1 - \mathrm { e } ^ { - \frac { 1 } { 3 } t } & t > 0 \end{cases}$$

1. Find the median delay and the 90th percentile delay.
2. Derive the probability density function of \(T\). Hence use calculus to find the mean delay.
3. Find the probability that a delay lasts longer than the mean delay. You are given that the variance of \(T\) is 9 .
4. Let \(\bar { T }\) denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of \(\bar { T }\).
5. A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling?

3 The time, in hours, until an electronic component fails is represented by the random variable \(X\). In this question two models for \(X\) are proposed.

1. In one model, \(X\) has cumulative distribution function $$\mathrm { G } ( x ) = \begin{cases} 0 & x \leqslant 0
   1 - \left( 1 + \frac { x } { 200 } \right) ^ { - 2 } & x > 0 \end{cases}$$ (A) Sketch \(\mathrm { G } ( x )\).
   (B) Find the interquartile range for this model. Hence show that a lifetime of more than 454 hours (to the nearest hour) would be classed as an outlier.
2. In the alternative model, \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 200 } \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0
   0 & \text { elsewhere. } \end{cases}$$ (A) For this model show that the cumulative distribution function of \(X\) is $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0
   1 - \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0 \end{cases}$$ (B) Show that \(\mathrm { P } ( X > 50 ) = \mathrm { e } ^ { - 0.25 }\).
   (C) It is observed that a particular component is still working after 400 hours. Find the conditional probability that it will still be working after a further 50 hours (i.e. after a total of 450 hours) given that it is still working after 400 hours.