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.
- 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. - 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.