Multiple independent components

8 A certain mechanical component has a lifetime, \(T\) months, which has a negative exponential distribution with mean 2.5.

1. A machine is fitted with 5 of these components which function independently.
   1. Find the probability that all 5 components are operating 3 months after being fitted.
   2. Find also the probability that exactly two components fail within one month of being fitted.
2. Show that the probability that \(n\) independent components are all operating \(c\) months after being fitted is equal to the probability that a single component is operating \(n c\) months after being fitted.

7 The lifetime, in hours, of a 'Trulite' light bulb is a random variable \(T\). The probability density function f of \(T\) is given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0
\lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \end{cases}$$ where \(\lambda\) is a positive constant. Given that the mean lifetime of Trulite bulbs is 2000 hours, find the probability that a randomly chosen Trulite bulb has a lifetime of at least 1000 hours. A particular light fitting has 6 randomly chosen Trulite bulbs. Find the probability that no more than one of these bulbs has a lifetime less than 1000 hours. By using new technology, the proportion of Trulite bulbs with very short lifetimes is to be reduced. Find the least value of the new mean lifetime that will ensure that the probability that a randomly chosen Trulite bulb has a lifetime of no more than 4 hours is less than 0.001 .

7 The time, \(T\) days, before an electrical component develops a fault has distribution function F given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \mathrm { e } ^ { - a t } & t \geqslant 0
0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. The mean value of \(T\) is 200 .

1. Write down the value of \(a\).
2. Find the probability that an electrical component of this type develops a fault in less than 150 days.
   A piece of equipment contains \(n\) of these components, which develop faults independently of each other. The probability that, after 150 days, at least one of the \(n\) components has not developed a fault is greater than 0.99 .
3. Find the smallest possible value of \(n\).