SimulationStatistics - Simulation Statistics Fundamentals A...

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Simulation Statistics Fundamentals A sample set is the set of all possible outcomes. Ex: The role of a dice produces random outcomes. The sample set is { } 6 , 5 , 4 , 3 , 2 , 1 = S . A distribution is the set of all probabilities in the sample set. The following are some restrictions: 1. S x x p i i 2200 0 ) ( 2. 1 ) ( = 2200 S x i i x p Example: For the dice we have. { } 6 , 5 , 4 , 3 , 2 , 1 6 1 ) ( 2200 = i i x x p A probability density function (PDF) is the continuos version of a distribution. It is a function that returns the probability when given a number in the sample set. The following are some restrictions: 1. S x x f i i 2200 0 ) ( 2. 1 ) ( = 2200 S x dx x f Example Let { } K , 6 , 5 , 4 , 3 , 2 , 1 = S and i i p 2 1 ) ( = then we can see that 1 2 1 ) ( 1 = = = 2200 k i i k S i Lim i p The Cumulative Density Function (CDF) is a function that returns the probability of the outcome being less than or equal to the input. = = = x k k p x X P x F 1 ) ( ) ( ) ( or in the continuos case
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- = = x dx x f x X P x F ) ( ) ( ) ( The mean is a measure of the tendency of the points in the population. The mean of a population is defined as -∞ = = x x xP ) ( m or in the continuos case dx x xf - = ) ( m The variance is a measure of the distance the points are from the mean. It is defined as -∞ = - = i i i x P x ) ( ) ( 2 2 m s or in the continuos case dx x f x - - = ) ( ) ( 2 2 m s A population is considered to be the complete set of points, i.e. the set of all people on earth. A sample is a subset of the population generally used to estimate parameters of the population. For a population = -∞ = = = = n i i x x n x xP mean 1 1 ) ( m and -∞ = -∞ = - = - = = x i i i i x n x P x 2 2 2 ) ( 1 ) ( ) ( var m m s And for a sample: = = = n i i x n x mean 1 1 and -∞ = - - = = x i x x n s 2 2 ) ( 1 1 var Note that x and 2 s are estimates of m and 2 s . The expected value of a sample set is the value that the argument of the expected value is expected to have. It is defined as -∞ = = x x P x g x g E ) ( ) ( )] ( [
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or in the continuos case dx x f x g x g E - = ) ( ) ( )] ( [ Note the expected value of a point in the population, ] [ x E , is the mean of the population and the expected value of the distance of a point in the population from the mean squared, ] ) [( 2 m - x E is the population's variance. Note that
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This note was uploaded on 09/13/2011 for the course EEL 5937 taught by Professor Staff during the Spring '08 term at University of Central Florida.

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SimulationStatistics - Simulation Statistics Fundamentals A...

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