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Unformatted text preview: Stat 5102 Lecture Slides Deck 1 Charles J. Geyer School of Statistics University of Minnesota 1 Empirical Distributions The empirical distribution associated with a vector of numbers x = ( x 1 ,...,x n ) is the probability distribution with expectation operator E n { g ( X ) } = 1 n n X i =1 g ( x i ) This is the same distribution that arises in finite population sam pling. Suppose we have a population of size n whose members have values x 1 , ... , x n of a particular measurement. The value of that measurement for a randomly drawn individual from this population has a probability distribution that is this empirical distribution. 2 The Mean of the Empirical Distribution In the special case where g ( x ) = x , we get the mean of the empirical distribution E n ( X ) = 1 n n X i =1 x i which is more commonly denoted x n . Those who have had another statistics course will recognize this as the formula of the population mean , if x 1 , ... , x n is considered a finite population from which we sample, or as the formula of the sample mean , if x 1 , ... , x n is considered a sample from a specified population. 3 The Variance of the Empirical Distribution The variance of any distribution is the expected squared deviation from the mean of that same distribution. The variance of the empirical distribution is var n ( X ) = E n n [ X E n ( X )] 2 o = E n n [ X x n ] 2 o = 1 n n X i =1 ( x i x n ) 2 The only oddity is the use of the notation x n rather than for the mean. 4 The Variance of the Empirical Distribution (cont.) As with any probability distribution we have var n ( X ) = E n ( X 2 ) E n ( X ) 2 or var n ( X ) = 1 n n X i =1 x 2 i  x 2 n 5 The Mean Square Error Formula More generally, we know that for any real number a and any random variable X having mean E { ( X a ) 2 } = var( X ) + (  a ) 2 and we called the lefthand side mse( a ), the mean square error of a as a prediction of X (5101 Slides 33 and 34, Deck 2). 6 The Mean Square Error Formula (cont.) The same holds for the empirical distribution E n { ( X a ) 2 } = var n ( X ) + ( x n a ) 2 7 Characterization of the Mean The mean square error formula shows that for any random vari able X the real number a that is the best prediction in the sense of minimizing the mean square error mse( a ) is a = . In short, the mean is the best prediction in the sense of mini mizing mean square error (5101 Slide 35, Deck 2). 8 Characterization of the Mean (cont.) The same applies to the empirical distribution. The real number a that minimizes E n { ( X a ) 2 } = 1 n n X i =1 ( x i a ) 2 is the mean of the empirical distribution x n . 9 Probability is a Special Case of Expectation For any random variable X and any set A Pr( X A ) = E { I A ( X ) } If P is the probability measure of the distribution of X , then P ( A ) = E { I A ( X ) } , for any event A (5101 Slide 63, Deck 1)....
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 Spring '03
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 Statistics, Probability

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