Probability Distributions - Download as PowerPoint by MrN305KI

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									          First 3 minutes
 Having read Anglesea Cabins and
Humboldt Oysters case consider the
  following common questions.
What
• is the marginal upside profit for a unit?
• is the marginal downside cost of a unit?
• things can you control?
• things can’t you control?
• concepts/elements of the course are applicable?
Theme: Coping with Uncertain Demand

                Anglesea Cabins   Humboldt Oysters

 Upside

 Downside

 Control

 Uncontrolled


 technique
Theme: Coping with Uncertain Demand
       Topics Covered Today

 • Anglesea Cabins Case
 • Normal distribution AWZ 6.2
 • Humboldt Oysters Case
               Anglesea Cabins

      The problem.


               Cnx
They call up
                  I decide


          Uncertain profit of $200 times x.
        Anglesea Cabins

Current policy.
• If C≤11 take n=C bookings
• If C>11, take n=11 bookings
• What is the chance of a “show”?
• X is binomial n=11, p=0.8

  Uncertain profit of $200 times x.
         Anglesea Cabins
When they take 11 bookings:
Profit =200  X
X has mean value np=110.8=8.8
Mean Profit =200  8.8 = $1760
  1760        20% loss         2200




What does the payoff table look like?
       Anglesea Cabins
When they take 11 bookings:
               TAKE n=11 BOOKINGS
       Shows(x)       Prob          Profit
          0                          $0
          1                         $200
          2                         $400
          3                         $600
          4                         $800
          5                         $1,000
          6                         $1,200
          7                         $1,400
          8                         $1,600
          9                         $1,800   Bino(11,0.8)
          10                        $2,000
          11                        $2,200
       Anglesea Cabins
When they take 11 bookings:
               TAKE n=11 BOOKINGS
       Shows(x)       Prob          Profit
          0          0.0000          $0
          1          0.0000         $200
          2          0.0000         $400
          3          0.0002         $600
          4          0.0017         $800
          5          0.0097         $1,000
          6          0.0388         $1,200
          7          0.1107         $1,400
          8          0.2215         $1,600
          9          0.2953         $1,800
          10         0.2362         $2,000
          11         0.0859         $2,200
                                    SIMULATE CASE

                Anglesea Cabins
When they take 12 bookings:
• What is the payoff table?
• What is the mean profit? How does this
compare to the previous profit?
• What is the chance of being under/over?
Have you gone far enough?
          31%


   1760           1899                  2200
                         20% loss




                                     anglesea_inclass.xls
         Anglesea Cabins
Why did I fix the number of calls C?
What about low demand seasons?
What if demand increases?
What about loss of customers?
What if Regal is full?
Other issues?
            Normal Distribution
The normal distribution
  • is used in market research, quality control
  (also in testing theory eg: GMAT)
  • describes error in many industrial processes
  • describes distribution of human features
  • is the basis of inferential statements based on
  regression output
  • is a continuous distribution

What is the probability of being 180cm tall?
            Normal Distribution
Continuous distributions described by a curve.
Areas under curve represent probabilities.
            Normal Distribution
The normal distribution defined by its
  mean μ – centre of the distribution
  standard deviation σ – spread of distribution
  Bell like shape


  calculus
  tables
  computer/XL
            Normal Distribution
Any normal variable is within
1 σ of μ 68%      of time
2 σ’s of μ 95%    of time
3 σ’s of μ 99.75% of time


  calculus
  tables
  computer/XL
      Standard Normal Distribution
The standard normal distribution is the normal
distribution with mean 0 and std. deviation 1
Non-standard normal variable can be made
standard normal variable by

                     X 
              Z
                        
Z measures how many std.dev’s above or below
the mean you are.
      Standard Normal Distribution
The standard normal distribution is the normal
distribution with mean 0 and std. deviation 1



                                  NORMSDIST(z)




                                    z
              Normal Distribution
                Computing Probabilities
Step 1: Convert question about variable X into a
question about standard normal variable Z.
Step 2: Draw a diagram* of the area you require
and express it in terms of areas to the left of z.

Step 3: Use NORMSDIST to obtain these areas.

                                             Java Demo



* The curve is symmetric about 0 and total area is 1.
              Normal Distribution
Graduate students sitting GMAT have scores that
are normally distributed with mean 570 and
standard deviation 60.
What is the chance someone scores over 700?

Let X be the GMAT score.
Normal (570,60)
Z = (X-570)/60 is standard normal.


  X  700  Z  700  570  / 60  2.1666
              Normal Distribution
Graduate students sitting GMAT have scores that
are normally distributed with mean 570 and
standard deviation 60.
What is the chance someone scores over 700?

Let X be the GMAT score.
Normal (570,60)
Z = (X-570)/60 is standard normal.


    Pr( X  700)  Pr( Z >2.166)
              Normal Distribution
Graduate students sitting GMAT have scores that
are normally distributed with mean 570 and
standard deviation 60.
What is the chance someone scores over 700?

Let X be the GMAT score.
Normal (570,60)
Z = (X-570)/60 is standard normal.
              Normal Distribution
Graduate students sitting GMAT have scores that
are normally distributed with mean 570 and
standard deviation 60.
What is the chance someone scores over 700?

Let X be the GMAT score.
Normal (570,60)
Z = (X-570)/60 is standard normal.
   Pr Z  2.166  1  NORMSDIST 2.166
                   1  0.9849
                   0.0151
             Normal Distribution
Graduate students sitting GMAT have scores that
are normally distributed with mean 570 and
standard deviation 60. What is the probability that
someone scores between 500 and 700?


X  500  Z  500  570  / 60  1.1666

 Pr(500  X  700)  Pr(1.167  Z  2.167)
          Normal Distribution




Pr(500  X  700)  Pr(1.167  Z  2.167)
                   Pr(Z  2.167)  Pr(Z  1.167)
               Normal Distribution
Graduate students sitting GMAT have scores that
are normally distributed with mean 570 and
standard deviation 60. What is the probability that
someone scores between 500 and 700?


Pr  1.167  Z  2.167   Pr Z  2.167   Pr Z  1.167 
          NORMSDIST 2.167   NORMSDIST  1.167 
          0.9849  0.1216  0.8633.
            Humboldt Oysters
• What is the distribution of unknown demand D?
What does it mean in simple language?
                     -$5.35     $2.45
            Humboldt Oysters
• What is the distribution of unknown demand D?
What if I set n=12150 production?
                    -$5.35     $2.45
                Humboldt Oysters
• What is the distribution of unknown demand D?
What if I set n=12150 production?
• What is probability that I sell it and make $2.45?
Use Excel and link so that you can change the
12150 policy.
• Calculate the mean profit on the last item.

      Outcome        Chance        Profit
         Sell                      $2.45
       Don’t sell                  -$5.35
                                    Class exercise
            Normal Distribution
Graduate students sitting GMAT have scores that
are normally distributed with mean 570 and
standard deviation 60. The top 1% of students
score over what GMAT score?


Use 2.33 for 1% percentiles.
Use 1.65 for 5% percentiles.
Use 1.28 for 10% percentiles.
Use 0.68 for 25% percentiles (quartiles).
                                  Class exercise
            Normal Distribution
Graduate students sitting GMAT have scores that
are normally distributed with mean 570 and
standard deviation 60. The top 1% of students
score over what GMAT score?


Normal variable is 2.33 std.dev’s above mean
with probability 1%.
For GMAT, 2.33 std.dev’s above mean is
               570+2.32660=709.6.
ANSWER is 710.
             Humboldt Oysters
Demand will exceed what value with probability
99%. What production level should I set to be
90% sure of selling everything?
 KEY TAKE AWAYS FROM CLASS
•In judging the level of production, two key quantities
 are the profit of selling the last unit and the loss
 incurred in not selling it.
•The normal distribution is a continuous distribution.
• Problems can always be translated to one involving
 standard normal probabilities.
• Excel:
          NORMSDIST(x) – prob less than x
          NORMSDIST(z) – for standard normal

								
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