text2_期望值与矩.pptVIP

Technical English For Communication Engineering;Expectations and Moments; Expectations, or expected values, are simply probabilistic averages of random variables (or functions of variables) in an experiment. We begin with the case of a single random variable, after which generalization to the multivariate case is simple.; Let g (X) be a function of a random variable X with a specified probability density function or probability mass（质量） function. Thus, Y = g (X) is another r. v. , which may be either discrete or continuous depending on the nature of the random variable X and the function g (X). The expected value of Y, written E [Y], is defined as (2.1a) ; In the case of discrete r. v. ’s, we replace the integral with a summation: (2.1b) The relation in (2.1) is sometimes called the fundamental theorem of expectation, but we take it as a definition. In words, the expected value is simply the probability-weighted average of values of g (X), and the notation E[ ] is merely shorthand for an integral operator.; 1. First and Second Moments Important special cases of this general definition are obtained when g (X) = X, the identity function, and when g (X) = X2. In the former case or (2.2) is called the first moment of X or, more commonly, the mean of X or the expected value of X.;For shorthand, the expected value of X will be represented by or occasionally by m when the random variable name is clear. To recall an analogy in mechanics, if the density function, or probability mass function, is symmetric about some value x0 , then = x0, provided the integral or sum in (2.2) exists. ; When g (X) = X2 , we have (2.3) which is called the second moment, or