4 15 points Planes arrive according to a Poisson process wit

4. (15 points) Planes arrive according to a Poisson process with a mean inter-arrival time of 5 minutes.

a. What is the expected number of planes arriving in a 4-hour period?

b. What is the probability that the next plane lands in 3 minutes or less?

c. What is the probability that exactly 13 planes land in the next hour?

d. What is the probability that 1 or less planes arrive in the next 10 minutes?

a. What is the expected number of planes arriving in a 4-hour period?

expected number=4*(60/5) = 48

---------------------------------------------------------------------------------------------------------------

b. What is the probability that the next plane lands in 3 minutes or less?

Given X~Poisson(mean=3/5=0.6 in 3 minutes)

P(X=x)=(0.6^x)*exp(-0.6)/x!

So P(X=1)=(0.6^1)*exp(-0.6)/1=0.329287

---------------------------------------------------------------------------------------------------------------

c. What is the probability that exactly 13 planes land in the next hour?

Given X~Poisson(mean=60/5=12 in a hour)

P(X=x)=(12^x)*exp(-12)/x!

So P(X=13) =(12^13)*exp(-12)/13! =0.1055704

---------------------------------------------------------------------------------------------------------------

d. What is the probability that 1 or less planes arrive in the next 10 minutes?

Given X~Poisson(mean=1*2=2 in 10 minutes)

P(X=x)=(2^x)*exp(-2)/x!

So P(X<=1)=P(X=0)+P(X=1)

=(2^0)*exp(-2)/1+(2^1)*exp(-2)/1

=0.4060058