## 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?

- 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))

- 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.

- 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?

**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.}

- 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?

**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