Question

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α = 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ = 8t. (Round your answers to three decimal places.)

(a) What is the probability that exactly 8 small aircraft arrive during a 1-hour period?

What is the probability that at least 8 small aircraft arrive during a 1-hour period?

What is the probability that at least 12 small aircraft arrive during a 1-hour period?

(b) What is the expected value and standard deviation of the number of small aircraft that arrive during a 105-min period?

(c) What is the probability that at least 23 small aircraft arrive during a 2.5-hour period?

What is the probability that at most 11 small aircraft arrive during a 2.5-hour period?

Answer 1

Let X be the number small aircraft arrive at a certain airport

X~ Poisson( 8 per hour)

a) Probability that exactly 8 small aircraft arrive during a 1-hour period

= P( X=8)

=

=0.1396

Probability that at least 8 small aircraft arrive during a 1-hour period

= P( X >=8)

= 1- P( X <8)

=1 - 0.45296

= 0.54704

Probability that at least 12 small aircraft arrive during a 1-hour period

= P( X >=12)

= 1- P( X <12)

=1 - 0.88807

= 0.11193

b) For 105-min period

X~ Poisson ( 8/60 per min)

X~ Poisson (( 0.133* 105) per 105 min)

X~ Poisson( 14 per 105 min)

Expected value= 14

Standard deviation= = =3.7416

c) For 2.5-hour period

X~ Poisson( 8*2.5 per 2.5 hour)

X~ Poisson( 20 per 2.5 hour)

Probability that at least 23 small aircraft arrive during a 2.5-hour period

= P( X>=23)

= 1- P( X < 23)

= 1- 0.72061

=0.27939

Probability that at most 11 small aircraft arrive during a 2.5-hour period

= P( X <=11)

= 0.02139