Explore the concepts of expectation, standard deviation, variance, and covariance. It is based on a lecture given by Professor Costis Maglaras at Columbia University. Inventory management problem. Variance Associates (VA) is a commercial aircraft leasing firm.
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Explore the concepts of expectation, standard deviation, variance, andcovariance
It is based on a lecture given by Professor Costis Maglaras at Columbia University.
Compute relative frequencies and tabulates the following probabilities:
FearUs Demand Oops Demand
Demand 0 1 2 3 4 Demand 0 1 2 3 4
Probability 0.2 0.1 0.4 0.1 0.2 Probability 0.1 0.2 0.4 0.2 0.1
0 1 2 3 4
0 0.02 0.01 0.04 0.01 0.02 0.1
Oops 1 0.04 0.02 0.08 0.02 0.04 0.2
Demand2 0.08 0.04 0.16 0.04 0.08 0.4
3 0.04 0.02 0.08 0.02 0.04 0.2
4 0.02 0.01 0.04 0.01 0.02 0.1
0.2 0.1 0.4 0.1 0.2
probability distribution for total demand:
Total Demand 0 1 2 3 4 5 6 7 8
Probability .02 .05 .14 .17 .24 .17 .14 .05 .02
Probability Distribution of Historical Monthly Cash Flow ($ in 000s)
Cash Flow ($600) ($450) ($300) ($150) $0 $150 $300 $450 $600
Probability 0.08 0.09 0.08 0.10 0.23 0.14 0.19 0.03 0.06
Cash Flow ($600) ($450) ($300) ($150) $0 $150 $300 $450 $600
E(Cash Flow) 0 0 0 0 0 0 0 0 0
Squared Dev 360,000 202,500 90,000 22,500 0 22,500 90,000 202,500 360,000
Sq Dev * Prob 28,800 18,225 7,200 2,250 0 3,150 17,100 6,075 21,600
FearUs Demand
0 1 2 3 4
0
1
2
3
4
Oops
Demand
Probability: the study of randomness
1.For any event A, 0 P(A) 1.
2.If A and B can never both occur (they are mutually exclusive), then
P(A and B) = P(A B) = 0.
3.P(A or B) = P(A B) = P(A) + P(B) - P(A B).
4.If A and B are mutually exclusive events, then P(A or B) = P(A B) = P(A) + P(B).
5.P(Ac) = 1 - P(A).
Independent Events
P(A and B) = P(A B) = P(A) P(B). (Markov??)
a) subscribes to both a free ISP and a paid ISP.
b) subscribes only to a paid ISP.
c) subscribes only to a free ISP.
P(A|B) = P(A holds given that B holds) = P(A∩B)/P(B)
Bayes
Laplace
PX(x) = P(X = x) = probability that the value of X is x.
The Normal Distribution
P(at least one empty seat) = = 1 - (0.9)16 = 0.815.
An 81.5% chance of having at least one empty seat! So the airline would be foolish not to overbook.
Reservation 20 19 18 17 16
E(Load) 15.943 15.839 15.599 15.132 14.396
E(Bumps) 2.057 1.261 0.600 0.167 0.000
E(Revenue) $1,183 $1,332 $1,440 $1,480 $1,440
E(X) = S xP(X=x)
where the summation is over all x that have P(X = x) > 0. It is sometimes denoted X or .
Var(X) = S (x- E(X))2P(X=x) ,
where the summation is over all x such that P(X = x) > 0. It is sometimes denoted 2(X) or .