moment properties

moment properties - Brigham Young University Department of...

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1 Brigham Young University Department of Economics Economics 459 - International Monetary Theory Useful Properties of Expected Values, Variances & Covariances The expected value, or mean, of a random variable, say x , is defined by, = dx x f x x E ) ( } { , where f ( x ) is the probability density function. The variance of a random variables is defined as, } }) { {( } { 2 x E x E x Var = , and the square root of this is called the standard deviation. The standard deviation can be interpreted as how far from the mean the variable is on average. Both the standard deviation and the variance must be positive. The covariance of two random variables, say x 1 & x 2 , is defined as, })} { })( { {( } , { 2 2 1 1 2 1 x E x x E x E x x Cov = . Note that if x 1 tends to be above its mean when x 2 is above its mean also, this will evaluate to a positive value. If x 1 tends to be below its means when x 2 is above its, then the covariance will be negative. Two random variables are said to be “independent” if their covariance is zero. Also note that if x 2 = x 1 , i.e. we consider the covariance of
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moment properties - Brigham Young University Department of...

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