Download MMPC Chapter 4

Chapters 9 – 14 Statistics Tutorial
and Introduction
© Holmes Miller 1999
Course theme:
Matching supply with demand
Course motivation:
Firms that are better at matching supply
with demand enjoy a significant
competitive advantage.
Now for a statistics tutorial to prepare you for
some material that lies ahead --
Normal distribution
 The density function for the normal
distribution looks like a
(symmetric) ‘bell-shaped’ curve
 For the standard normal distribution, the mean (m) is 0 and the SD (s) is 1
 Concerning the AREA under the curve, about
68% is within 1 SD of the mean
95% is within 2 SDs
99.7% is within 3 SDs
(AREA is the proportion of observations in an interval)
Density and Cumulative Distribution Functions
For the normal distribution
For Demand F(Q) is Prob {Demand <= Q)
Normal distribution
 Standard Normal
 General Normal
Mean = 0
Std dev = 1
2
-1
0
1
2
m - 2s
m-s
m
m+s
m + 2s
Sample Problems
 The annual precipitation amounts for Allentown are normally distributed
with a mean of 32 in. and a standard deviation of 5 in.
 If one year is randomly selected, find the probability that the mean
precipitation in a year is less than 29 in? Less than 40in? Greater than
40 in?
 Use two methods
 Standardize using formula z = (X – m) / s and use the Excel function:
NORMSDIST(x)
 Use the Excel function NORMDIST(X, m, s, 1)
 Answers:
 less than 29 in? 0.274
 Less than 40in? 0.945
 Greater than 40 in? 0.055
Expected Value
 The expected value of something happening is the “payoff” from each
possible outcome multiplied by the probability of that outcome summed
over all of the outcomes
 Mathematically if v(i) is the payoff if event i occurs, and if p(i) is the
probability that it occurs, the expected value is: Si = p(i) * v(i)
 Example: From a deck of cards, if you draw a heart you win $100 and if
you draw a diamond you win $70. If you draw a spade you win $50 and if
you draw a club you lose $200. What is your expected payoff?
 Answer: (.25)*100 + (.25)*70 +(.25)*50 + (.25)(-200) = $5
Expected Value Problem
Probabilities
Payoffs
Alt 1
Alt
Alt 3
Event A
0.25
100
75
145
Event B
0.45
200
250
275
Event C
0.30
300
275
180
Alt 1
Expected return
205
Alt 2
213.75
Alt 3
214
Another Game Situation
You have won $64,000 and are facing a tough $125,000 question.
You have four choices and are guessing so there is a 25% chance
you will guess right and a 75% chance you will guess wrong.
This means that you will receive $125,000 if you answer the
question correctly. But you will receive only $32,000 if your
answer is wrong. Also, you can walk away with $64,000 if you
decide not to answer the question.
What would be the best strategy to take?
Loss Function
Rolling Dice
 The Loss Function L(Q) is the
expected amount that X is
greater than Q
 Example: When rolling dice, Loss
Function for rolling more than a
seven
 If X is normally distributed, we
can use loss function in appendix
B of the text
(a)
(b)
Roll
Prob
Amount X exceeds 7
(a) * (b)
2
0.0278
0
0
3
0.0556
0
0
4
0.0833
0
0
5
0.1111
0
0
6
0.1389
0
0
7
0.1667
0
0
8
0.1389
1
0.139
9
0.1111
2
0.222
10
0.0833
3
0.250
11
0.0556
4
0.222
12
0.0278
5
0.139
L(7) = Total of last column
0.972
Loss Function Problems
 Normal Distribution
 Loss Function for -2.21? (2.2147)
 Loss function for 1.83? (0.0132)
 Loss Function for 2.70?
 Loss Function for -0.058?