Question

At the Canyon Marina in San Antonio, it has been reported that during summer working days, according to the Poisson distribution, boats arrive at a rate of 3 per hour.

Arrival

per hour

Arrival

per minute

λ

boats in 20 minute period

a. What is the expect number of boats that will arrived in a 20-minute period?

boats

x

P(x)

b. What is the probability that no boats arrive during a 20-minute period?

boats

c. What is the probability that 2 boats arrive during a 20-minute period?

boats

d. What is the probability that 6 boats arrive during a 20-minute period?

boats

e. What is the probability that more than 7 boats arrive during a 20-min. period?

boats

Answer 1

let X denotes the number of boats arrive at the Canyon Marina in San Antonio in 20 min duration.

now according to Poisson distribution boats arrive at a rate of 3 per hour

that is 3 boats in 60 minutes. so in 20 minutes 3/60*20=1 boat arrive.

so X follows Poisson Distribution with parameter 1

so pmf of X is P[X=x]=e^-11^x/x! x=0,1,2,3...............

a) so expected number of boats arrive is

E[X]=

=

=e^-1*e=1  

hence expected number of boats arrive=1 [answer]

b) probability that no boats arrive in 20 minute period=P[X=0]=e^-11⁰/0!=e^-1=0.367879 [answer]

c) probability that 2 boats will arrive in 20 minute period=P[X=2]=e^-11²/2!=0.18393972 [answer]

d) probability that 6 boats will arrive in 20 minute period=P[X=6]=e^-11⁶/6!=0.000510943 [answer]

e) probability that more than 7 boats will arrive in 20 minute period=P[X>7]

=1-P[X<=7]=1-P[X=0]-P[X=1]-P[X=2]-P[X=3]-P[X=4]-P[X=5]-P[X=6]-P[X=7]=1-e^-1/0!-e^-1/1!-e^-1/2!-e^-1/3!-e^-1/4!-e^-1/5!-e^-1/6!-e^-1/7!

=1-e^-1(1+1+1/2+1/3!+1/4!+1/5!+1/6!+1/7!)=0.000010249 [answer]