[理學(xué)]概率論與數(shù)理統(tǒng)計(jì)英文第三章-資料下載頁
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【正文】 es, then ()(d) If , then .(e) If and , then .(f) For any constant , ()(g) Schwarz’s inequality. （許瓦慈不等式）Let X,Y be random variables, then ()The equality holds iff or for some constant a.Proof (a) By the definition..In the summation, for the term and for the other terms thus a.(b), (c) The proofs are given in advanced probability theory, so is omitted here.(d) If , then in the summation each term is nonnegative, thus .(e).If and , then in the summation the left side is and each term in right side is nonnegative, so each term is . Thus, for the terms , we must have . This means .(f).Set in (b), we have (h) Assume , then . Consider the variable , s is a constant, then put we haveIf then , the equality holds in (). If , then so the equality holds. On the other hand, if the equality holds in (), but , then we must have , 2．Variance 方差Except the expectation of a random variable, we are interested in some other quantities related to a random variable. Let’s consider an example.ExampleA cigarette manufacturer tests tobaccos grown from two districts for nicotine contents, obtains the following resultsDistrict 1 24, 27, 25, 22, 22 (in milligrams)District 2 28 ,27, 25, 20, 20The average nicotine content for both district are the same： 24 milligrams. But the manufacturer prefer the tobaccos from district 1, because it has smaller dispersion than district 2, ., it is more stable.To measure the dispersion of a data set , whose average is , we use the quantity “variance”, denote by , is defined asFor example, if and are the variances of nicotine constant for the district 1and 2, resp., then.For many purpose it is desirable that a measure of dispersion be expressed in the same unit as the original data, thus the square root of the variance, called standard deviation is used. Thus for the data set , the standard deviation （標(biāo)準(zhǔn)差）is To measure the dispersion of random variables, we also use the quantity variance and standard deviation.Definition Let be a discrete random variable, having expectation . Then the variance of , denote by is defined as the expectation of the random variable ()The square root of the variance , denote by , is called the standard deviation of : ()The variance of random variable has the following properties.（方差性質(zhì)）Theorem For a discrete random variable ,(a) ()(b) , are constants. ()(c ) If , then , . is a constant.Proof (a) Let (b) Thus (c) Since , by Theorem (e) , . Example A die is tossed. Find the variance and standard deviation of the spots shown, if(a)this die is a fair die, . the probability distribution of is, (b)the probability distribution of isX123456P(X=k) Solution (a) By Example . . Thus (b)Example Find the variance of the amount won by the player in Example .（）Solution We have , and. 泊松分布的期望和方差Theorem The expectation and variance of a Poisson random variable with parameter are, respectively, and . ()Proof By the definition, . Variance of Binomial distribution52
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