Exponential Distribution:

The exponential distribution is used to model the time between the occurrence of events in an interval of time. The time elapsed before an earthquake strikes in a region? Time, we need to wait before a customer arrives our shop? How long will it take before a call center receives the next phone call? How long will a piece of machinery work without breaking down?

  1. The time spends by one in a bank is exponentially distributed with mean 10 minutes, λ = 1/10. What is the probability that a customer will spend more than 15 minutes in the bank? What is the probability that a customer will spend more than 15 minutes in the bank given that he is still in the bank after 10 minutes?

    Solution: P(X > 15) = e-15λ= e−3/2=0.22
    P(X > 15|X > 10) = P(X > 5) = e−1/2 = 0.604
    Failure rate (hazard rate) function r(t) r(t) = f(t) / (1 − F(t))

  2. People immigrate into a territory at a Poisson rate λ = 1 per day.
    (a) What is the expected time until the tenth immigrant arrives?
    (b) What is the probability that the elapsed time between the tenth and the eleventh arrival exceeds 2 days?
    Solution:
    ime until the 10th immigrant arrives is S10.
    E(S10) = 10/λ = 10.
    P (T11 > 2) = e−2λ = 0.133.

  3. If immigrants to area A arrive at a Poisson rate of 10 per week, and if each immigrant is of English descent with probability 1/12, then what is the probability that no people of English descent will immigrate to area A during the month of February?

  4. Solution: The number of English descent immigrants arrived up to time t is N1(t),
    which is a Poisson process with mean λ/12 = 10/12.

    P(N1(4) = 0) = e−(λ/12) •4= e−10/3.

  5. On a road, motorcycle pass according to a Poisson process with rate 5 per minute. buses pass according to a Poisson process with rate 1 per minute. The two processes are independent. If in 3 minutes, 10 vehicles passed by. What is the probability that 2 of them are buses?

  6. Solution: Each vehicle is independently a car with probability 5 5+1 = 5 6 and a truck with probability 1 6. The probability that 2 out of 10 vehicles are buses is given by the binomial distribution:

    Solution :
    = 10C2*(1/6)2*(5/8)8