1. Chapters 9 – 14 Statistics Tutorial and Introduction © Holmes Miller 1999

2. Matching supply with demand Course theme: 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 --

3. Normal distribution The density function for the normal distribution looks like a (symmetric) ‘bell-shaped’ curve For the standard normal distribution, the mean (  ) is 0 and the SD (  ) 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)

4. Density and Cumulative Distribution Functions For the normal distribution For Demand F(Q) is Prob {Demand <= Q)

5. Normal distribution Standard Normal General Normal  - 2   -    +   + 2  2 -1 0 1 2 Mean = 0 Std dev = 1

6. 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 –  /  and use the Excel function: NORMSDIST(x)  Use the Excel function NORMDIST(X,  ) äAnswers: äless than 29 in? 0.274 äLess than 40in? 0.945 äGreater than 40 in? 0.055

7. 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:  i = 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

8. Expected Value Problem Alt 1Alt 2Alt 3 Expected return205213.75214

9. 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? Another Game Situation

10. Loss Function 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 Rolling Dice (a)(b) RollProbAmount X exceeds 7(a) * (b) 20.027800 30.055600 40.083300 50.111100 60.138900 70.166700 80.138910.139 90.111120.222 100.083330.250 110.055640.222 120.027850.139 L(7) = Total of last column0.972

11. Loss Function Problems Normal Distribution äLoss Function for -2.21? (2.2147) äLoss function for 1.83? (0.0132) äLoss Function for 2.70? äLoss Function for -0.058?