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Statistics and Data Analysis

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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. Binomial and Poisson Distributions Counting (Discrete) Distributions Discrete uniform Bernoulli Distribution Binomial Counting Distributions and Probability Models Poisson Process and Poisson Distribution

    4. Building Block 1 Discrete Uniform Distribution X takes M equally likely values. May be “coded” various ways (spots on a die, shopping malls chosen by a shopper, cards in a deck, etc.) Probability of each outcome = 1/M Dots on a die; P(x) = 1/6, x = 1,2,3,4,5,6 52 cards; P(x) = 1/52, x = 1,…,52, some numbering 0/1 Coin toss; P(x) = ½, x = 0,1 For compound events, count the outcomes; P(Diamond) = P(A?)+P(K?)+…+P(2 ?) = 1/52 + 1/52 + … + 1/52 = 1/4

    5. Building Block 2 Bernoulli Random Variable X = 0 or 1 Probabilities: P(X = 1) = ? P(X = 0) = 1 – ? (X = 0 or 1 corresponds to an event occurring or not occurring)

    6. Binomial Probability P(r successes in R trials) = number of ways r successes can occur in n trials times the probability of r successes times the probability of (R-r) failures r = Sum of R Bernoulli trials; r = Si xi P(r successes in R trials) =

    7. Other Distributions Based on Counting Geometric: Probability of R trials before the first success occurs. Hypergeometric: Probability of mixtures of more than one kind of success. E.g., a hand in bridge has 13 cards. What is the probability of 5 Spades, 2 Hearts, 3 Clubs, 3 Diamonds?

    8. Counting Rules If trials are independent, with constant success probability ?, then uniform, Bernoulli and binomial distributions give the exact probabilities of the outcomes. They are counting rules. The “assumptions” are met in reality.

    9. Models Settings in which the probabilities can only be approximated Models “describe” reality but don’t match it exactly Assumptions are descriptive Outcomes are not limited to a finite range

    10. Extreme Binomials Success probability, ?, gets very small. Number of trials, R, becomes very large R? stays fairly constant ? ? Many common settings isolated in space or time: Phone calls that arrive at a switch per second. Customers that arrive at a service point per minute Number of bomb craters per square kilometer during WWII in London Number of accidents per hour at a given location Number of buy orders per minute for a certain stock Number of individuals who have a disease in a large population Number of plants of a given species per square kilometer In principle, x, the number of occurrences, could be huge (essentially unlimited) in every case.

    11. Poisson Model The Poisson distribution is a model that fits situations such as these very well.

    12. Poisson Model for Counts

    13. Poisson Variable

    14. Application: Major Derogatory Reports

    15. Doctor visits by people in a sample of 27,326

    16. Diabetes Incidence per 1000

    17. Disease Incidence

    18. Poisson Distribution of Disease: Cases in 1000 Draws

    19. Calc->Probability Distributions->Poisson

    20. Application The arrival rate of customers at a bank is 3.2 per hour. What is the probability of 6 customers in a particular hour?

    21. Scaling The mean can be scaled up to the appropriate time unit or area Ex. Arrival rate is 3.2/hour. What is the probability of 9 customers in 2 hours? The arrival rate will be 6.4 customers per 2 hours, so we use Prob[X=9|?=6.4] = 0.0824844.

    22. Application: Hospital Beds Cardiac care unit handles heart attack victims on the day of the incident. In the population served, heart attacks are Poisson with mean 4.1 per day If there are 5 beds in the unit, what is the probability of an overload?

    23. Application – Poisson Arrivals

    24. Application: Peak Loading (Peak Loading Problem) If they have 7 beds, the expected vacancy rate is 7 - 4.1 = 2.9 beds, or 2.9/7 = 42% of capacity. This is costly. (This principle applies to any similar operation with random demand, such as an electric utility.) They must plan capacity for the peak demand, and have excess capacity most of the time. A business tradeoff found throughout the economy.

    25. An Economy of Scale Suppose the arrival rate doubles to 8.2. The same computations show that the hospital does not need to double the size of the unit to achieve the same 90% adequacy. Now they need 12 beds, not 14. The vacancy rate is now (12-8.2)/8.2 = 32%. Better. The hospital that serves the larger demand has a cost advantage over the smaller one.

    26. Summary Basic building blocks Uniform (equally probable outcomes) Set of independent Bernoulli trials Counting Distribution: Binomial Poisson Model Poisson processes The Poisson distribution for counts of events

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