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Statistics and Data Analysis

Statistics and Data Analysis. . Part 9

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Statistics and Data Analysis

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    1. Statistics and Data Analysis Professor William Greene Stern School of Business IOMS Department Department of Economics

    2. Statistics and Data Analysis

    3. The Normal Distribution Continuous Distributions Application – The Exponential Model Computing Probabilities Normal Distribution Model Normal Probabilities Reading the Normal Table Computing Normal Probabilities Applications

    4. Continuous Distributions Continuous distributions are models for probabilities of events associated with measurements rather than counts. Continuous distributions do not occur in nature the way that discrete ones (e.g., binomial) do. The continuous distribution describes nature with a model. The random variable is a measurement, x The device is a probability density function, f(x). Probabilities are computed using calculus (and computers)

    5. Application: Light Bulb Lifetimes A box of light bulbs states “Average life is 2000 hours” P[Fails at exactly 2000 hours] is 0.0. Note, this is exactly 2000.000000000, not 2000.0000000001, … P[Fails in an interval (1500 to 2500)] is provided by the model (as we now develop). The model being used is the exponential model,

    6. Model for Light Bulb Lifetimes

    7. Model for Light Bulb Lifetimes

    8. Continuous Distribution

    9. Probability of a Single Value

    10. Difference of CDFs

    11. Computing a Probability

    12. Probability Computation

    13. Applications of the Exponential Model Other uses for the exponential model: Time between signals arriving at a switch (telephone, message center,…) (This is called the “interarrival time.”) Length of survival of transplant patients. (Survival time) Many others More elaborate models build onto the exponential model.

    14. Normal Distribution The most useful distribution in all branches of statistics and econometrics. Strikingly accurate model for elements of human behavior and interaction Strikingly accurate model for any random outcome that comes about as a sum of small influences.

    15. Applications Biological measurements of all sorts (not just human mental and physical) Accumulated errors in experiments Numbers of events accumulated in time Amount of rainfall per interval Number of stock orders per (longer) interval. (We used the Poisson for short intervals) Economic aggregates of small terms. And on and on…..

    16. The Normal Distribution

    18. Demoivre’s Result

    19. Normal Density – The Model

    20. Normal Densities

    21. SAT Scores with Mean 500, Standard Deviation 100

    22. The Empirical Rule and the Normal Distribution

    23. Computing Probabilities P[x = a specific value] = 0. (Always) P[a < x < b] = P[x < b] – P[x < a] (Note, for continuous distributions, < and < are the same because of the first point above.)

    24. Textbooks Provide Tables of Areas for the Standard Normal

    25. Computing Probabilities Standard Normal Tables give probabilities when µ = 0 and s = 1. For other cases, do we need another table? Probabilities for other cases are obtained by “standardizing.” Standardized variable is z = (x – µ)/ s z has mean 0 and standard deviation 1

    26. “Standard” Normal Density

    27. Standard Normal Density

    28. Standard Normal Distribution Facts The random variable z runs from -8 to +8 f(z) > 0 for all z, but for |z| > 4, it is essentially 0. The total area under the curve equals 1.0. The curve is symmetric around 0. (The normal distribution generally is symmetric around µ.)

    29. Only Half the Table Is Needed

    30. Only Half the Table Is Needed

    31. Areas Left of Negative Z

    32. Compute Normal Probabilities

    33. Computing Probabilities by Standardizing: Example

    34. Compute Normal Probabilities If SAT scores are scaled to have a normal distribution with mean 500 and standard deviation 100, what proportion of students would be expected to score between 450 and 600?

    35. Modern Computer Programs Make the Tables Irrelevant

    36. Application of Normal Probabilities

    37. A Normal Probability Problem The amount of cash demanded in a bank each day is normally distributed with mean $10M (million) and standard deviation $3.5M. If they keep $15M on hand, what is the probability that they will run out of money for the customers? Let $X = the demand. The question asks for the Probability that $X will exceed $15M.

    38. Inverse Normal Probability For the bank in the previous exercise, how much cash must they keep on hand to be 99% certain that they will have enough cash to meet the daily demand? We need to find the $CM such that P[$X > $CM] < .01. This requires us to use the normal table “backwards.”

    39. Inverse Normal

    40. Summary Continuous Distributions Models of reality The density function Computing probabilities as differences of cumulative probabilities Application to light bulb lifetimes Normal Distribution Background Density function depends on µ and s The empirical rule Standard normal distribution Computing normal probabilities with tables and tools

    41. Additional applications and exercises, see “Notes on the Normal Distribution,” especially pages 1-11.

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