Link Poisson to exponential

10 The number of hits per minute on a particular website has a Poisson distribution with mean 0.8. The time between successive hits is denoted by \(T\) minutes. Show that \(\mathrm { P } ( T > t ) = \mathrm { e } ^ { - 0.8 t }\) and hence show that \(T\) has a negative exponential distribution. Using a suitable approximation, which should be justified, find the probability that the time interval between the 1st hit and the 51st hit exceeds one hour.

8 The number of flaws in a randomly chosen 100 metre length of ribbon is modelled by a Poisson distribution with mean 1.6. The random variable \(X\) metres is the distance between two successive flaws. Show that the distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.016 x } & x \geqslant 0 ,
0 & x < 0 , \end{cases}$$ and deduce that \(X\) has a negative exponential distribution, stating its mean. Find

1. the median distance between two successive flaws,
2. the probability that there is a distance of at least 50 metres between two successive flaws.

8 The number of flaws in a randomly chosen 100 metre length of ribbon is modelled by a Poisson distribution with mean 1.6. The random variable \(X\) metres is the distance between two successive flaws. Show that the distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.016 x } & x \geqslant 0
0 & x < 0 \end{cases}$$ and deduce that \(X\) has a negative exponential distribution, stating its mean. Find

1. the median distance between two successive flaws,
2. the probability that there is a distance of at least 50 metres between two successive flaws.

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.

1. Explain why, for \(x \geqslant 0 , \mathrm { P } ( X > x ) = \mathrm { e } ^ { - 0.8 x }\).
2. Find the probability that there is at least one flaw in a 4 metre length of cloth.
3. Find
   (a) the distribution function of \(X\),
   (b) the probability density function of \(X\),
   (c) the interquartile range of \(X\).

6. Customers arrive at a shop such that the number of arrivals in a time interval of \(t\) minutes follows a Poisson distribution with mean \(0.5 t\).

1. Find the probability that exactly 5 customers arrive between 11 a.m. and 11.15 a.m.
2. A customer arrives at exactly 11 a.m.
   1. Let the next customer arrive at \(T\) minutes past 11 a.m. Show that $$P ( T > t ) = \mathrm { e } ^ { - 0.5 t }$$
   2. Hence find the probability density function, \(f ( t )\), of \(T\).
   3. Hence, giving a reason, write down the mean and the standard deviation of the time between the arrivals of successive customers.