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Unformatted text preview: 1 PPF501 PART 2 DISTRIBUTIONS AND STATISTICAL INFERENCE 2 To describe the outcome of a probabilistic or statistical process, we define a random variable a numerical equivalent to a group of events 3 EXAMPLE: We make 4 investments (S = success, F = failure) Let X = # of successes X = 0 equivalent to FFFF X = 1 equivalent to FFFS FFSF FSFF SFFF 4 Probability distribution A (mutually exclusive and collectively exhaustive) list of the possible random variable values , and the probability associated with each possible value . 5 Example : You invest in four investment opportunities, each with a 50/50 chance of success, each opportunity independent of each other opportunity. Let X = number of successes. What are the possibilities for X, the number of successes you get? Answer: 0, 1, 2, 3, or 4. 6 What is the probability of each possibility? X f(X) = probability of X 0 .0625 1 .2500 2 .3750 3 .2500 4 .0625 1.0000 7 PARAMETERS E(X) = Expected Value of X is the long run average value of X. (That is, repeat the experiment many many times, and average the results of what comes out. This is E(X)) 8 Example: X f(X) Xf(X) 0 .0625 0 1 .2500 .2500 2 .3750 .7500 3 .2500 .7500 4 .0625 .2500 2.000 = E(X) [Mathematically, E(X) = i (X i )f(X i )] 9 Another Example: X f(X) Xf(X) .2 0 1 .5 .5 2 .3 .6 1.1= E(X) NOTE: E(X) is not necessarily one of the possible values of X 10 (X) = Standard Deviation of X is a measure of the variability of X. (That is, how far away from the center, E(X), will individual values of X tend to be?) [Mathematically, (X) = SQRT{ i (X i E(X) ) 2 f(X i )}] 11 Example: X f(X) [XE(X)] 2 [XE(X)] 2 f(X) 0 .0625 (2) 2 = 4 .2500 1 .2500 (1) 2 = 1 .2500 2 .3750 0 3 .2500 (+1) 2 = 1 .2500 4 .0625 (+2) 2 = 4 .2500 2 = 1 and = SQRT ( 2 ) = SQRT (1) = 1 12 BINOMIAL DISTRIBUTION 1. The Experiment consists of n TRIALS 2. Each trial has TWO and ONLY TWO outcomes SUCCESS and FAILURE 3. Each trial is independent of each other trial 4. P(SUCCESS) = the same for each trial = p 5. Experimenter wishes to know X, the number of successes out of the n trials We know 0 X n and X is an integer. 13 For given values of n = number of trials and p = probability of success on any one trial, we typically seek to find the probability that we get X successes out of the n trials: f(X  n , p ) 14 There is a mathematical formula for f(X  n , p ), f(X  n , p ) = { n !/(X![ nX]!)} p X (1 p ) nX but one hardly ever uses it, because we can determine the answer easily using EXCEL. 15 The major EXCEL commands available: BINOMDIST(X, n , p , 0) BINOMDIST(X, n , p , 1). 16 So, given the values of n and p , BINOMDIST(X, n , p , 0) = P(result exactly = X) BINOMDIST(X, n , p , 1) = P(result is X) So, if n = 10, and p = .1, P(result = 2) = BINOMDIST(2, 10, .1, 0) = .19371 =BINOMDIST(2, 10, .1, 0) 0.19371 P(result 2) = BINOMDIST(2, 10, .1, 1) = .929809 =BINOMDIST(2, 10, .1, 1) 0.929809 17...
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 Spring '11
 KEATING

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