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PMF of flight booking

wtrow

New member
Joined
Jan 24, 2011
Messages
17
An airline company wants to book reservations for its popular flight at 7.00 AM from Detroit
to New York. There are 100 openings for passengers. A person with a confirmed ticket will
not show up for the flight with probability 0.08. For each passenger (person with a booking
and who shows up), the airline makes a profit of 75 Dollars. The airline overbooks the flight.
Let N denote the number of bookings for the flight. Note that N >= 100. Treat N as a
parameter that the airline would like to choose. Let K denote the number of persons (among
N) with confirmed ticket who show up for the flight. If K is greater than 100, (K ? 100)
people are randomly selected, and will be forced to travel on the next flight, and each of them
is given a coupon of 100 Dollars. In other words, the airline makes a loss of 100 Dollars for
each such person. Let D denote the profit that airline makes for each flight.

(a) For a fixed parameter N, find the PMFs of random variables K and D.
(b) For a fixed parameter N, find the expected value of D.
(c) Find N that will maximize the expected profit.

So I am stuck on part A, and I feel like I need to know that one before I advance. I just don't know how to actually make a PMF for variables that can be chosen at random.
 

tkhunny

Moderator
Staff member
Joined
Apr 12, 2005
Messages
9,708
Have you considered:

q = 0.08
p = 1 - 0.08 = 0.92

\(\displaystyle P(K=i) = \frac{N!}{(i!)\cdot (N-i)!}\cdot q^{i}\cdot p^{N-i}\) for \(\displaystyle i \in 0,1,2...,N\)
 

wtrow

New member
Joined
Jan 24, 2011
Messages
17
Thank you for the reply. That makes sense. Now for finding D, the profit, I think it would just be:

P(k=i)*75 for i E 0,1,2...99
P(k=i)*75 - P(k=i)*100 for i E 100,101,...N

Or in other words, the PMF of K times D. Is this right or did I mess up again?
 

tkhunny

Moderator
Staff member
Joined
Apr 12, 2005
Messages
9,708
1) It doesn't seem to matter wich one containes i = 100, does it?
2) We're looking for a PMF. Simply multiplying by 75 or 100 violates the premise of summing to unity. You'll have to adjust for that.
 
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