The random variable \(T\) is the lifetime, in hours, of a particular type of battery. It is given that \(T\) has a negative exponential distribution with mean 500 hours.
(i) Write down the probability density function of \(T\).

(ii) Find the probability that a randomly chosen battery of this type has a lifetime of more than 750 hours.

(iii) Find the median value of \(T\).

The time, \(T\) days, before an electrical component develops a fault has distribution function F given by
\[\mathrm{F}(t)=\left\{\begin{array}{ll}
1-\mathrm{e}^{-a t} & t \geqslant 0 \\
0 & \text { otherwise }
\end{array}\right.\]
where \(a\) is a positive constant. The mean value of \(T\) is 200 .
(i) Write down the value of \(a\).
(ii) 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 .
(iii) Find the smallest possible value of \(n\).

The random variable \(T\) is the lifetime, in hours, of a randomly chosen battery of a particular type. It is given that \(T\) has a negative exponential distribution with mean 400 hours.
(i) Write down the probability density function of \(T\).

(ii) Find the probability that a battery of this type has a lifetime that is less than 500 hours.
(iii) Find the median of the distribution.

As shown in the diagram, the continuous random variable \(X\) has probability density function f given by
\(f(x)=\left\{\begin{array}{ll} m x & 0 \leqslant x \leqslant 2, \\ \frac{k}{x^{2}}+c & 2 \leqslant x \leqslant 6, \\ 0 & \text { otherwise, } \end{array}\right.\)
where \(m, k\) and \(c\) are constants.
(a) Given that \(\mathrm{P}(X \leqslant 2)=\frac{1}{3}\), show that \(m=\frac{1}{6}\) and find the values of \(k\) and \(c\).

(b) Find the exact numerical value of the interquartile range of \(X\).

The continuous random variable \(X\) has probability density function \(f\) given by
\(\mathrm{f}(x)=\left\{\begin{array}{ll} k & 0 \leqslant x\lt 1 \\ k x & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array}\right.\)
where \(k\) is a constant.
(a) Show that \(k=\frac{2}{5}\).
(b) Find the interquartile range of \(X\).

(c) Find \(\operatorname{Var}(X)\).