2003 Paper 3 Q12

Year: 2003
Paper: 3
Question Number: 12

Course: UFM Statistics
Section: Poisson Distribution

Difficulty: 1700.0 Banger: 1470.9

Poisson distribution exponential distribution probability density function geometric distribution conditional expectation inter-arrival times waiting time expected value

Problem

Brief interruptions to my work occur on average every ten minutes and the number of interruptions in any given time period has a Poisson distribution. Given that an interruption has just occurred, find the probability that I will have less than \(t\) minutes to work before the next interruption. If the random variable \(T\) is the time I have to work before the next interruption, find the probability density function of \(T\,\). I need an uninterrupted half hour to finish an important paper. Show that the expected number of interruptions before my first uninterrupted period of half an hour or more is \(\e^3-1\). Find also the expected length of time between interruptions that are less than half an hour apart. Hence write down the expected wait before my first uninterrupted period of half an hour or more.

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1470.9

Banger Comparisons: 2

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Problem source

Brief interruptions to my work occur on average every ten minutes 
and the number of interruptions in any given time period has a Poisson distribution. 
Given that an interruption has just occurred, find the probability 
that I will have less than $t$ minutes to work before the next interruption. 
If the random variable $T$ is the time I have to work before the next interruption, 
find the probability density function of $T\,$.   
      
I need an uninterrupted half hour to finish an important paper.  
Show that the expected number of interruptions before my first 
uninterrupted period of half an hour or more is $\e^3-1$. 
Find also the expected length of time between interruptions 
that are less than half an hour apart.
Hence write down 
the expected wait before my first uninterrupted period of half an hour or more.

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