数学期望论文文献

数学期望Mathematical ExpectationIn probability theory the expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff). Thus, it represents the average amount one "expects" as the outcome of the random trial when identical odds are repeated many times. Note that the value itself may not be expected in the general sense - the "expected value" itself may be unlikely or even expected value from the roll of an ordinary six-sided die is , which is not among the possible outcomes:A common application of expected value is to gambling. For example, an American roulette wheel has 38 places where the ball may land, all equally likely. A winning bet on a single number pays 35-to-1, meaning that the original stake is not lost, and 35 times that amount is won, so you receive 36 times what you've bet. Considering all 38 possible outcomes, the expected value of the profit resulting from a dollar bet on a single number is the sum of what you may lose times the odds of losing and what you will win times the odds of winning:The change in your financial holdings is −$1 when you lose, and $35 when you win. Thus one may expect, on average, to lose about five cents for every dollar bet, and the expected value of a one-dollar bet is $. In gambling, an event of which the expected value equals the stake (of which the bettor's expected profit is zero) is called a "fair game."[edit] Mathematical definitionIn general, if is a random variable defined on a probability space (where Ω is the sample space and F is the cumulative distribution function of probability, ()), then the expected value of (denoted or sometimes or ) is defined aswhere the Lebesgue integral is employed. Note that not all random variables have an expected value, since the integral may not exist (., Cauchy distribution). Two variables with the same probability distribution will have the same expected value, if it is in the gambling example mentioned the probability distribution of X admits a probability density function f(x), then the expected value can be computed asIt follows directly from the discrete case definition that if X is a constant random variable, . X = b for some fixed real number b, then the expected value of X is also expected value of an arbitrary function of X, g(X), with respect to the probability density function f(x) is given by:[edit] Conventional terminologyWhen one speaks of the "expected price", "expected height", etc. one means the expected value of a random variable that is a price, a height, etc. When one speaks of the "expected number of attempts needed to get one successful attempt," one might conservatively approximate it as the reciprocal of the probability of success for such an attempt. Cf. expected value of the geometric distribution. [edit] Properties[edit] ConstantsThe expected value of a constant is equal to the constant itself; ., if 'c' is a constant, then E(c) = c[edit] MonotonicityIf X and Y are random variables so that almost surely, then .[edit] LinearityThe expected value operator (or expectation operator) is linear in the sense thatCombining the results from previous three equations, we can see that -for any two random variables X and Y (which need to be defined on the same probability space) and any real numbers a and b.[edit] Iterated expectation[edit] Iterated expectation for discrete random variablesFor any two discrete random variables X,Y one may define the conditional expectation:which means that is a function on the expectation of X satisfiesHence, the following equation holds:The right hand side of this equation is referred to as the iterated expectation and is also sometimes called the tower rule. This proposition is treated in law of total expectation.[edit] Iterated expectation for continuous random variablesIn the continuous case, the results are completely analogous. The definition of conditional expectation would use inequalities, density functions, and integrals to replace equalities, mass functions, and summations, respectively. However, the main result still holds:[edit] InequalityIf a random variable X is always less than or equal to another random variable Y, the expectation of X is less than or equal to that of Y:If , then .In particular, since and , the absolute value of expectation of a random variable is less than or equal to the expectation of its absolute value:[edit] RepresentationThe following formula holds for any nonnegative real-valued random variable X (such that ), and positive real number α:In particular, this reduces to:[edit] Non-multiplicativityIn general, the expected value operator is not multiplicative, . is not necessarily equal to . If multiplicativity occurs, the X and Y variables are said to be uncorrelated (independent variables are a notable case of uncorrelated variables). The lack of multiplicativity gives rise to study of covariance and correlation.[edit] Functional non-invarianceIn general, the expectation operator and functions of random variables do not commute; that isA notable inequality concerning this topic is Jensen's inequality, involving expected values of convex (or concave) functions.[edit] Uses and applications of the expected valueThe expected values of the powers of X are called the moments of X; the moments about the mean of X are expected values of powers of . The moments of some random variables can be used to specify their distributions, via their moment generating empirically estimate the expected value of a random variable, one repeatedly measures observations of the variable and computes the arithmetic mean of the results. If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the residuals (the sum of the squared differences between the observations and the estimate). The law of large numbers demonstrates (under fairly mild conditions) that, as the size of the sample gets larger, the variance of this estimate gets classical mechanics, the center of mass is an analogous concept to expectation. For example, suppose X is a discrete random variable with values xi and corresponding probabilities pi. Now consider a weightless rod on which are placed weights, at locations xi along the rod and having masses pi (whose sum is one). The point at which the rod balances is .Expected values can also be used to compute the variance, by means of the computational formula for the varianceA very important application of the expectation value is in the field of quantum mechanics. The expectation value of a quantum mechanical operator operating on a quantum state vector is written as . The uncertainty in can be calculated using the formula .[edit] Expectation of matricesIf X is an matrix, then the expected value of the matrix is defined as the matrix of expected values:This is utilized in covariance matrices.[edit] ComputationIt is often useful to update a computed expected value as new data comes in. This can be done as follows, where new_value is the count-th value, and we use the previous estimate to compute :[edit] Formula for non-negative integral valuesWhen a random variable takes only values in {0,1,2,3,...} we can use the following formula for computing its expectation:For example, suppose we toss a coin where the probability of heads is p. How many tosses can we expect until the first heads? Let X be this number. Note that we are counting only the tails and not the heads which ends the experiment; in particular, we can have X = 0. The expectation of X may be computed by . This is because the number of tosses is at least i exactly when the first i tosses yielded tails. This matches the expectation of a random variable with an Exponential distribution. We used the formula for Geometric progression: .

概率统计是研究随机现象与统计规律的学科，数学期望是反映随机变量总体取值的平均水平的一个数字特征。虽然随机变量的概率分布能完整地描述随机变量的统计规律，但是在实际问题中，要获得随机变量的概率分布不是一件简单的事情，所以我们往往要知道一些从某些方面刻画随机变量特征的数值，从而也可以清晰地解决实际问题。数学期望则完美地演绎了这一角色。这篇论文主要介绍了数学期望的来源，定义，性质以及应用。让我们更加深刻地认识数学期望应用的广泛性以及对于分析实际问题的重要性。

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参考文献是毕业论文中的一个重要构成部分,它的引用是对论文进行引文统计和分析的重要信息来源。下文是我为大家搜集整理的关于数学论文参考文献的内容，欢迎大家阅读参考!数学论文参考文献(一) [1]李秉德，李定仁，《教学论》，人民教育出版社，1991。 [2]吴文侃，《比较教学论》，人民教育出版社，1999 [3]罗增儒，李文铭，《数学教学论》，陕西师范大学出版社，2003。 [4]张奠宙，李士 ，《数学教育学导论》高等教育出版社，2003。 [5]罗小伟，《中学数学教学论》，广西民族出版社，2000。 [6]徐斌艳，《数学教育展望》，华东师范大学出版社，2001。 [7]唐瑞芬，朱成杰，《数学教学理论选讲》，华东师范大学出版社，2001。 [8]李玉琪，《中学数学教学与实践研究》，高等教育出版社，2001。 [9]中华人民共和国教育部制订，《全日制义务教育数学课程标准(实验稿)》,北京：北京师范大出版社,2001. [10] 高中数学课程标准研制组编，《普通高中数学课程标准》，北京：北京师范大出版社,2003. [11]教育部基础教育司，数学课程标准研制组编，《全日制义务教育数学课程标准解读(实验稿)》,北京：北京师范大出版社,2002. [12]教育部基础教育司组织编写，《走进新课程——与课程实施者对话》，北京：北京师范大出版社,2002. [13]新课程实施过程中培训问题研究课题组编，《新课程与学生发展》，北京：北京师范大出版社,2001. 数学论文参考文献(二) [1]新课程实施过程中培训问题研究课题组编，《新课程理念与创新》，北京：北京师范大出版社,2001. [2][苏]AA斯托利亚尔，《数学教育学》，北京：人民教育出版社，1985年。 [3][苏]斯涅普坎，《数学教学心理学》，时勘译，重庆：重庆出版社，1987年。 [4]张奠宙，《数学教育研究导引》，南京：江苏教育出版社，1998年。 [5]丁尔升，《中学数学教材教法总论》，北京：高等教育出版社，1990年。 [6]马忠林，等，《数学教育史简编》，南宁：广西教育出版社，1991年。 [7]魏群，等，《中国中学数学教学课程教材演变史料》，北京：人民教育出版 社，1996年。 [8]张奠宙，等，《数学教育学》，南昌：江西教育出版社，1991年。 [9]严士健，《面向21世纪的中国数学教育》，南京：江苏教育出版社，1994年。 [10]傅海伦，《数学教育发展概论》,北京：科学出版社，2001年。 [11]李求来，等，《中学数学教学论》，长沙：湖南师范大学出版社，1992年。 [12]章士藻，《中学数学教育学》，南京：江苏教育出版社，1996年。 [13]十三院校协编组，《中学数学教材教法》，北京：高等教育出版社，1988年。 [14][美]美国国家研究委员会，方企勤等译，《人人关心数学教育的未来》，北 京：世界图书出版公司，1993年。 [15]潘菽，《教育心理学》，北京：人民教育出版社，1980年。 数学论文参考文献(三) [1]孙艳蕊，张祥德.利用极小割计算随机流网络可靠度的一种算法[J],系统工程学报，2010,25(2)，284-288. [2]孔繁甲，王光兴.基于容斥原理与不交和公式的一个计算网络可靠性方法,电子学报，1998,26(11)，117-119. [3]王芳，侯朝侦.一种计算随机流网络可靠性的新算法[J],通信学报，2004,25(1)，70-77. [4][J],Networks,1987,17(2)：227-240. [5]],(1)：46-49. [6][J],(4)：325-334. [7](3)：389-395. [8]. [9]封国林，鸿兴，魏凤英.区域气候自忆预测模式的计算方案及其结果m.应ni气象学报，1999,10:470. [10]达朝究.一个可能提高GRAPES模式业务预报能力的方案[D].兰州：兰州人学，2011 [11]符综斌，干强.气候突变的定义和检测方法[j].大气科学，1992,16(4)：482-492. [12]顾震潮.天数值预报屮过去资料的使用问题[J].气象学报，1958,29:176. [13]顾震潮.作为初但问题的天气形势数值预报由地而天气历史演变作预报的等值性[J].气象学报，1958,29:93. [14]黄建平，H纪范.海气锅合系统相似韵现象的研究[J].中NI科学(B)，1989,9:1001. [15]黄建平，王绍武.相似-动力模式的季节预报试验[J].国科学(B)1991,21:216. 猜你喜欢： 1. 统计学论文参考文献 2. 关于数学文化的论文免费参考 3. 关于数学文化的论文优秀范文 4. 13年到15年参考文献论文格式 5. 浅谈大学数学论文范文

1、离散随机变量的一切可能值与对应的概率P的乘积之和称为数学期望,记为E若随机变量ξ仅取值x1,x2,x3,.,xn,其概率分别为p1,p2,p3,.,pn,称加权平均值p1x1+p2x2+p3x3+.+pnxn,为随机变量ξ的数学期望,通常记为Eξ.。