You are traveling from San Francisco to Casper, Wyoming, with a change of planes

Question

You are traveling from San Francisco to Casper, Wyoming, with a change of planes in Denver. Your arrival time in Denver is uniformly distributed between 12:00 noon and 1:00 PM. Your connecting flight leaves Denver between 12:30 and 1:30 PM and the departure time is also uniformly distributed and independent of the arrival time.

(a) Assuming you require 20 minutes to change planes, what is the probability that you will make the connection?

(b) Given that your arrival is before 12:30 PM, what is the probability that you will make the connection? (Remember, you still need 20 minutes to change planes.)

(c) What is the average amount of time you must wait in Denver? (Waiting does not include the 20 minutes needed to make the connection and, if you miss the flight, assume your waiting
time is zero.)

(d) Repeat parts (a), (b), and (c) under the assumption that your arrival time has some distribution given by a density function fa (which is nonzero only between noon and 1) and the departure time has density function fd (nonzero only between 12:30 and 1:30). In other words, make sure you can set up the appropriate integrals (with the correct limits of integration) to calculate the probabilities requested in parts (a), (b), and (c).

Explanation / Answer

Arrival time in Denver is uniformly distributed between 12:00 noon and 1:00 PM.

Departure time of flight is uniformly distributed between 12:30 noon and 1:30 PM.

Change tome = 20min

a) probability of connecting planes

It is possible when next flights departs 20 min later when u land from previous plane.

from 12:00 to 12:10 whenever he lands he can catch, if departing flight arrives after 1:20 he can catch any time

So, the compulsary catching of flight is 1/10*1/10=1/100

b) Given that your arrival is before 12:30 PM,  the probability that you will make the connection

he can catch the flight only after 12:50

So, the time remaining is 1:30-12:50= 40 min

probablity of catching next flight when arrived at 12:30 is 40/60=2/3

for before 12:30= arrival probability * catching probability

=1/2* 2/3= 1/3

c) Average waiting time

minimum waiting time = 0

maximun waiting time= 1:30-12:20 = 70 min

Average waiting time

= (max+min)/2

=35

_____________________________________________________________

As per rules of chegg 3 sub parts can be limited to answer

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