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Spring 2010 STAT230 midterm2 Package final_merged-1

Spring 2010 STAT230 midterm2 Package final_merged-1 - STAT...

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STAT 230 Midterm 2 Review Package Spring 2010 1 Spring 2010 STAT 230 Midterm 2 Review Package Waterloo SOS Prepared by Grace Gu
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STAT 230 Midterm 2 Review Package Spring 2010 2 Table of Contents Important formulas (Memorizing these should help!) ............................................ 3 Chapter 5 Discrete Distributions .......................................................................... 4 Chapter 7 Expectation, Averages and Variability ............................................... 11 Chapter 8 Discrete Multivariate Distributions ................................................... 14 Extra Practice ....................................................................................................... 20 Past Midterm 2 ………………………………………………………………………….……………………….28
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STAT 230 Midterm 2 Review Package Spring 2010 3 Important formulas (Memorizing these should help!) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. If X and Y are independent , then Cov(X, Y) = 0 12. The correlation coefficient of X and Y is 13. 14. 15. If we have n identically distributed random variables, and a i = 1 for all I = 1, …, n 16.
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STAT 230 Midterm 2 Review Package Spring 2010 4 Chapter 5 Discrete Distributions Definitions Random Variable - a function that assigns a real number to each point in a sample space S. Probability function (p.f.) of a discrete random variable X - the function Cumulative distribution function (cdf) of a random variable X - the function 1. The following are the properties of a cdf F(x): a. b. c. 2. Distributions A) Discrete Uniform Distribution. If X takes on values a, a+1, a+2, . . . , b with all values being equally likely , then X has a discrete uniform distribution on [a, b]. B) Hypergeometric Distribution We pick n objects at random without replacement from a collection of N items, and X is the number of successes among the n objects picked. Then, X has a hypergeometric distribution. Intuition:
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STAT 230 Midterm 2 Review Package Spring 2010 5 Numerator: We use the counting techniques from Chapter 3. We have r “success items” within the collection of N items. We select x objects out of the r objects, and select the remaining n-x objects out of the N- r “failure items”. Denominator: If we don’t impose any restrictions, we can choose all n objects from any of the N items available. C) Binomial Distribution Suppose we conduct an experiment that results in Success, and Failure (a Bernoulli r.v.). Let the probability of success be p and the probability of failure be 1-p. We then repeat the experiment n independent times. Let X be the number of successes obtained. Then X has a binomial distribution . Intuition: The x successes can happen in any of the n trials, and the x successes and n-x failures are repeats. Thus, by the counting techniques from chapter 3, we can arrange them in ways. Since each one of the n trials is independent, by the multiplication rule from chapter 4, we can simply multiply all the probabilities together. D) Geometric Distribution Suppose we conduct an experiment that can either result in success (with probability p) or failure (with probability 1-p). We keep repeating the experiment independently until we obtain a success . Let X be the number of failures obtained before the first success. Then X has a geometric distribution.
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