Question

The number of actual fire emergencies (emergencies where there is actually a fire) per week as the fire service
in Bergen can be described with a Poisson distribution with parameter λ = 1.8.

a)

What is the probability of exactly three calls in a week?
What is the probability of more than three calls in a week? What is the probability of at least an emergency during one day?
What is the expected number of calls in a year?

The time, measured in number of days, that goes between two subsequent real fire events is exponentially distributed with parameter λ = 1.8 / 7 = 0.26 (the time between subsequent events in a Poisson process is exponentially distributed).

b)

What is the expected number of days between two consecutive calls?
What is the probability that there will be more than 1 day between two subsequent calls?
What is the probability that less than 2 days will elapse between two subsequent calls?
What is the probability that there will be between 1 and 2 days between two subsequent calls?

Answer 1

Let X be a random variable denoting number of actual fire emergencies per week.

XPOISSON(1.8)

PMF of X is given by:

for x=0,1,2,3,...............

a)   

• P(Exactly 3 calls in a week)=
  
• P(more than 3 calls in a week)=
  
• Let Y be a R.V. denoting the number of fire emergencies per day.
  The expected number of fire emergencies in 7 days is 1.8.
  The expected number of fire emergencies in 1 days is 0.26
  Clearly,Y POISSON(0.26)
  

  PMF of Y is given by:

  for y=0,1,2,3,...............

• 1 Year= 52 Weeks
  Expected number of calls in a year
  X_1,X_2,X_{3,...........}X₅₂Poisson(1.8) i.i.d
  
  
  
  

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