CHAPTER 2 – DISCRETE DISTRIBUTIONS HœSEYIN GœLER MATHEMATICAL...

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Transcript of CHAPTER 2 – DISCRETE DISTRIBUTIONS HœSEYIN GœLER MATHEMATICAL...

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CHAPTER 2 DISCRETE DISTRIBUTIONS HSEYIN GLER MATHEMATICAL STATISTICS Discrete Distributions 1 Slide 2 2.1. DISCRETE PROBABILITY DISTRIBUTIONS The concept of random variable: S: Space or support of an experiment A random variable (r.v.) X is a real valued function defined on the space. X: S R x: Represents the value of X x S X is a discrete r.v. if its possible values are finite, or countably infinite. Discrete Distributions 2 Slide 3 A chip is selected randomly from the bowl: Discrete Distributions 3 S = {1, 2, 3, 4} X: The number on the selected chip X is a r.v. with space S x = 1, 2, 3, 4. X is a discrete r.v. (it takes 4 different values) Slide 4 P(X = x): Represents the probability that X is equal to x. Discrete Distributions 4 The distribution of probability on the support S The probability mass function (p.m.f.) Slide 5 Discrete Distributions 5 Slide 6 CALCULATING PROBABILITIES USING P.M.F. Discrete Distributions 6 Compute the probability that the number on the chip is 3 or 4. If A is a subset of S then Slide 7 CALCULATING PROBABILITIES USING P.M.F. Discrete Distributions 7 Compute the probability that the number on the chip is less than or equal to 3. Slide 8 RELATIVE FREQUENCIES AND RELATIVE FREQUENCY HISTOGRAM Discrete Distributions 8 The histogram of relative frequencies is called relative frequency histogram. Relative frequencies converge to the p.m.f as n increases. When the experiment is performed n times the relative frequency of x is Slide 9 The chip experiment is repeated n = 1000 times using a computer simulation. Discrete Distributions 9 x1234Total Frequency 98209305388 n = 1000 h (x) 0.0980.2090.3050.388 1 Slide 10 THE COMPARISON OF f(x) AND h(x) Discrete Distributions 10 f(x) is theoretically obtained while h(x) is obtained from a sample. x1234Total f (x) 0.10.20.30.4 1 h (x) 0.0980.2090.3050.388 1 Slide 11 THE MEAN OF THE (PROBABILITY) DISTRIBUTION Discrete Distributions 11 called the mean of X. It is possible to estimate using relative frequencies. The weighted average of X is Slide 12 THE MEAN OF THE EMPIRICAL DISTRIBUTION x 1, x 2,..., x n : Observed values of x f j : The frequency of u j u j = 1, 2, 3, 4. Discrete Distributions 12 the empirical distribution the mean of the empirical distribution or the sample mean Slide 13 THE VARIANCE AND THE STANDARD DEVIATION OF THE DISTRIBUTION Discrete Distributions 13 The variance of X is The standart deviation of X is Slide 14 AN ALTERNATIVE FOR THE VARIANCE OF THE DISTRIBUTION Discrete Distributions 14 r_th moment about the origin Slide 15 THE VARIANCE OF THE EMPIRICAL DISTRIBUTION Discrete Distributions 15 Slide 16 THE VARIANCE AND THE STANDART DEVIATION OF THE SAMPLE Discrete Distributions 16 s 2 (the variance of the sample) is an estimate of (the variance of X).