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 .

It is known that the average lifetime of hair dryers from a certain manufacturer is 2 years. The lifetimes are exponentially distributed.

1. Find the probability that the lifetime of a randomly selected hair dryer is between 1·8 and 2·5 years. [4]
2. Given that 20% of hair dryers have a lifetime of at least \(k\) years, find the value of \(k\). [3]
3. Jon buys his first hair dryer from the manufacturer today. He will replace his hair dryer with another from the same manufacturer immediately when it stops working. Find the probability that, in the next 5 years, Jon will have to replace more than 3 hair dryers. [3]
4. State one assumption that you have made in part (c). [1]