9 Cotton cloth is sold from long rolls of cloth. The number of flaws on a randomly chosen piece of cloth of length \(a\) metres has a Poisson distribution with mean \(0.8 a\). The random variable \(X\) is the length of cloth, in metres, between two successive flaws.
- Explain why, for \(x \geqslant 0 , \mathrm { P } ( X > x ) = \mathrm { e } ^ { - 0.8 x }\).
- Find the probability that there is at least one flaw in a 4 metre length of cloth.
- Find
(a) the distribution function of \(X\),
(b) the probability density function of \(X\),
(c) the interquartile range of \(X\).