# PMF of flight booking

#### wtrow

##### New member
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
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
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
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.