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. 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

solution:-

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

XPOISSON(1.8)

PMF of X is given by:-

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...

• 1year = 52 weeks
• expected number of calls in ayear
• X_1,X_2,X_{3,...........}X₅₂Poisson(1.8) i.i.d

​​​​​​

  

thank u

please give me thumb up......