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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) X●f(X) 0 .0625 0 1 .2500 .2500 2 .3750 .7500 3 .2500 .7500 4 .0625 .2500 2.000 = E(X) [Mathematically, E(X) = Σ i (X i )●f(X i )] 9 Another Example: X f(X) X●f(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 σ (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, σ (X) = SQRT{ Σ 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 σ 2 = 1 and σ = SQRT ( σ 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
 Normal Distribution, Probability, Standard Deviation

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