Question

The number of airplane landings at a small airport follows a Poisson distribution with a mean rate of 3 landings every hour.

4.1 (6%)

Compute the probability that 3 airplanes arrive in a given hour?

4.2 (7%)

What is the probability that 8 airplanes arrive in three hours?

4.3 (7%)

What is the probability that more than 3 airplanes arrive in a period of two hours?

Answer 1

Let X denotes the number of airplane landings at a small airport in 1 hour.

X Poisson( = 3)

Pr[X=3] = e ^-3  / 3! = 0.224041807 = 0.2240 ( correct up to 4 decimal places )

   The probability that 3 airplanes arrive in a given hour is 0.224041807

Let Y denotes the number of airplane landings at a small airport in 3 hours.

The number of occurrences in the time interval of length 1 hour follows a Poisson distribution with mean 3.

Then, the number of occurrences in the time interval of length 3 hours follows a Poisson distribution with mean 9.

Y Poisson( = 9)

Pr[Y=8] = e ^-8  / 8! = 0.13175564 = 0.1317 ( correct up to 4 decimal places )

   The probability that 8 airplanes arrive in 3 hours is 0.13175564

Let Z denotes the number of airplane landings in a period of 2 years.

Z   Poisson( = 6)

Pr[ Z > 3] = 1 - (Pr[Z = 0] + Pr[ Z = 1] + Pr[Z =2] + Pr[Z =3])

= 1 - ( e ^-0  / 0! +    e ^-1  / 1! +  e ^-2  / 2! +  e ^-3  / 3! )

= 1 - (0.002478752 + 0.014872513 + 0.044617539 + 0.089235078)

= 0.848796118

= 0.8488 ( correct up to 4 decimal places )

So, the probability that more than 3 planes arrive in a period of 2 hours is 0.8488.