Find threshold for given probability

7 The random variable \(T\) is the lifetime, in hours, of a randomly chosen decorative light bulb of a particular type. It is given that \(T\) has a negative exponential distribution with mean 1000 hours.

1. Write down the probability density function of \(T\).
2. Find the probability that a randomly chosen bulb of this type has a lifetime of more than 2000 hours. A display uses 10 randomly chosen bulbs of this type, and they are all switched on simultaneously. Find the greatest value of \(t\) such that the probability that they are all alight at time \(t\) hours is at least 0.9 .

5 The distance, \(X \mathrm {~km}\), completed by a new car before any mechanical fault occurs has distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - a x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. The mean value of \(X\) is 10000 . Find

1. the value of \(a\),
2. the probability that a new car completes less than 15000 km before any mechanical fault occurs. The probability that a new car completes at least \(d \mathrm {~km}\) before any mechanical fault occurs is 0.75 .
3. Find the value of \(d\).

6 The distance, \(X\) metres, between successive breaks in a water pipe is modelled by an exponential distribution. The mean of \(X\) is 25 The distance between two successive breaks is measured. A water pipe is given a 'Red' rating if the distance is less than \(d\) metres. The government has introduced a new law changing \(d\) to 2
Before the government introduced the new law, the probability that a water pipe is given a 'Red' rating was 0.05 6

1. Explain whether or not the probability that a water pipe is given a 'Red' rating has increased as a result of the new law.
   6
2. Find the probability density function of the random variable \(X\). 6
3. After investigation, the distances between successive breaks in water pipes are found to have a standard deviation of 5 metres. Explain whether or not the use of an exponential model in parts (a) and (b) is appropriate.
   [0pt] [2 marks]

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

1. Write down the value of \(a\). [1]
2. Find the probability that an electrical component of this type develops a fault in less than 150 days. [2]

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.

1. Find the smallest possible value of \(n\). [4]