Understanding Random Variables: Probability, Moments, and Distributions - Prof. 1435, Apuntes de Psicología

¡Descarga Understanding Random Variables: Probability, Moments, and Distributions - Prof. 1435 y más Apuntes en PDF de Psicología solo en Docsity! Theme 5: Random variables Moments (Op “Técnicas de investigación” O Universitat de Barcelona Academic course 2012-2013 * Solanas, A., Salafranca, Ll., Fauquet, J. y Núñez, M. I. (2005). Estadística descriptiva en ciencias del comportamiento. Madrid: Thomson. Where: 4/2 EST What: Capítulo 3 *A random variable is a correspondence between: a) events that can occur when a random experiences is taking place, e.g., the results when tossing a coin. b) real numbers, which are infinite. *Example: a coin is tossed 10 times and the frequency of heads is tallied. Suppose the results is 7. A correspondence has been made between the result of a random experience and a real number obtained counting the amount of heads. *A random variable requires knowing the probabilities of each of the possible results!!! (e.g., a student answering at random). Or perform an empirical study (prob of a number of correct answers not at rrandom?) *When a coin is tossed 10 times, different results (number of heads) can occur. Given that the results can be 0 or 10 heads or anything in between, each experience can be associated with a real number. *This application between results and real numbers is usually not bijective, i.e., not every distinct result of the random experience has a corresponding different real number. That is, there are different ways in which 7 heads can be obtained in 10 tosses. *There is a single sequence of results leading to 0 heads, but there are different sequences leading to 5 heads. Thus, it is not as common to obtain 0 heads as it is to obtain 5 heads in 10 tosses. *So the random variable “number of heads” requires a quantification of the degree of certainty we have for each of the possible results. *Probability is the concept used for quantifying the differential frequency of each result of the random experience. Result of the random experience  corresponding real number  probability assigned. Event prob.,Trials 0,2,25 Binomial Distribution x pr ob ab ili ty 0 5 10 15 20 25 0 0,04 0,08 0,12 0,16 0,2 Event prob. 0,5 Geometric Distribution x pr ob ab ili ty 0 2 4 6 8 10 0 0,1 0,2 0,3 0,4 0,5 Event prob.,Successes 0,45,10 Negative Binomial Distribution x pr ob ab ili ty 0 10 20 30 40 50 0 0,02 0,04 0,06 0,08 Mean 3 Poisson Distribution x pr ob ab ili ty 0 2 4 6 8 10 12 0 0,04 0,08 0,12 0,16 0,2 0,24 What’s in between? *Density: informs about the amount of values in an interval. Density ≠ probability. *The probability is obtained for an interval of values centered at x *The probability of an individual value is assumed to be zero. / 2 / 2 ( ) ( ) x dx x dx p x f x dx    Mean,Std. dev. 0,1 Normal Distribution -5 -3 -1 1 3 5 x 0 0,1 0,2 0,3 0,4 de ns ity Mean 10 Exponential Distribution x de ns ity 0 10 20 30 40 50 60 0 0,02 0,04 0,06 0,08 0,1 Lower limit,Upper limit 0,2 Uniform Distribution x de ns ity 0 0,4 0,8 1,2 1,6 2 0 0,1 0,2 0,3 0,4 0,5 Shape 10 Pareto Distribution x de ns ity 1 1,5 2 2,5 3 3,5 4 0 2 4 6 8 10 Mean,Std. dev. 0,1 Normal Distribution -5 -3 -1 1 3 5 x 0 0,2 0,4 0,6 0,8 1 cu m u la tiv e p ro b a b ili ty Mean 10 Exponential Distribution x cu m u la tiv e p ro b a b ili ty 0 10 20 30 40 50 60 0 0,2 0,4 0,6 0,8 1 Lower limit,Upper limit 0,2 Uniform Distribution x cu m u la tiv e p ro b a b ili ty 0 0,4 0,8 1,2 1,6 2 0 0,2 0,4 0,6 0,8 1 Shape 10 Pareto Distribution x cu m u la tiv e p ro b a b ili ty 1 1,5 2 2,5 3 3,5 4 0 0,2 0,4 0,6 0,8 1 *The moments of a random variable are global indicators of some of its characteristics, unlike mass probability, density and distribution function which give information about each of the values/intervals. *Other global indicators: based on position (quantiles). Mathematical expectancy: information about location. Variance: information about scatter. Skewness: information about shape; equally distant from the mean lower and higher values? Kurtosis: information about shape; as peaky as the Normal distribution? More? Less? *A k-order non-centered moment is defined for discrete random variables as *A k-order moment centered with respect to the arbitrary point c is defined for discrete random variables as *In both expressions n designates the amount of values of the random variable.     1 Prob n kk k i i i E X x xm X           ' 1 Prob n k k k i i i E X c X x x c       *4 questions: true/false  Math expectancy (Correct=2) *0 correct (also all correct): 1 result  *1 correct (also all but one correct): 4 results     *2 correct: 6 results       *Mathematical expectancy: the most probable individual value *All possible results: 1 + 4 + 6 + 4 + 1 = 16 Prob (C=0) = 1/16 = 0.0625 Prob (C=1) = 4/16 = 0.25 Prob (C=2) = 6/16 = 0.375 Prob (C=3) = 4/16 = 0.25 Prob (C=4) = 1/16 = 0.0625 E(C) = 0.0625*0 + 0.25*1 + … + 0.0625*4 = 2 *Any other value is more probable!!! *6 questions: true/false  Math expectancy (Correct=3) *Mathematical expectancy: the most probable individual value *All possible results: 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64 Prob (C=0) = Prob (C=6) = 1/64 = 0.0156 Prob (C=1) = Prob (C=5) = 6/64 = 0.0938 Prob (C=2) = Prob (C=4) = 15/64 = 0.2344 Prob (C=3) = 20/64 = 0.3125 E(C) = 0.0156*0 + 0.0938*1 + … + 0.0156*6 = 3 *Any other value is more probable!!! *More than three correct is more probable!!! *The E is less probable than before!!! *Variance is the second-order moment centered with respect to the mathematical expectancy. *For discrete random variables it is defined as *Variance is an indicator of the grouping or scatter of the values of the random variable. Its square root is the “standard deviation”: same measurement unit as the variable.       2 22 2 1 Prob n i i i E X X x x          *Properties: If X is a random variable, 2 = Var(X)  0. If a is a constant, Var(a) = 0. If a (scale) and b (location) are constants, Var(aX + b) = a2 Var(X): unaffected by changes in location. Var(X) < E[(X – a)2], for every a  E(X): the reason for choosing the reference point. *Grades of two students in an exam scored from 0 to 10. *Variance = 0.  If a is a constant, Var(a) = 0; minimum. Maximum =? *Grades of three students in an exam scored from 0 to 10. *Variance = 2.666. Meaning? *Grades of two students in an exam scored from 0 to 100. *Variance = 100. Meaning? * If a (10) and b (0) are constants, Var(aX + b) = a2 Var(X). *Grades of four students in an exam scored from 0 to 100; there two identical grades, twice. *Variance = 100. Meaning? * Kurtosis coefficient: quantifies the degree to which a mass probability or a density function is flat or peaky. Defined as 4th order moment centered with respect to E(X). in order to make nondimensional. So that for the normal distribution the coefficient would yield 0 (mesokurtic). If the distribution is symmetric and unimodal, a positive value denotes a leptokurtic distribution and a negative value a platykuritc. Unaffected by changes in location and scale. 4 2 4 3      * Skewness = 0 * Kurtosis = −0.3 (Normal ≠ triangular) * Skewness = 0 * Kurtosis = 0.8 (more peaky) * Skewness = 0 * Kurtosis = −1.3 Most platykurtic? * Skewness = 0 * Kurtosis = −1.8 * Skewness = 0.7 * Kurtosis = −0.8 not comparable *Coefficient of variation: quantifies scatter with no unit of measurements  comparison between variables in different metrics. *If the random variable can take negative values, CV is defined as *Sometimes multiplied by 100. CV    CV    *Theoretical, ideal representations of reality based on testable assumptions. *The models are used for describing how a variable behaves empirically. *A model is chosen according to: Whether the variables is discrete or continuous. The likelihood of the assumptions. *Possible: no ideal model matches sufficiently well the empirical facts.