概率论与数理统计(英文)

Probability theory and mathematical statistics 概率论与数理统计 Continuous Random Variable 连续型随机变量 Continuous random variables appear when we deal will quantities that are measured on a continuous scale. For instance, when we measure the speed of a car, the amount of alcohol in a person's blood, the tensile strength of new alloy. Continuous Random Variable 1. Definition Definition 4.1.1 A function f(x) defined on (, ) is called a probability density function （概率密度函数）if: (i) f(x)0 for any xR; (ii) f(x) is intergrable on (, )and f(x)dx1. Definition 4.1.2 Let f(x) be a probability density function. If X is a random variable having distribution function xF(x)P(Xx)f(t)dt, (4.1.1) then X is called a continuous random variable having density function f(x). In this case, x2P(x1Xx2)f(t)dt. (4.1.2) x1 2. Example Example 4.1.2 Find k so that the following can serve as the probability density of a continuous random variable: f(x)k (x). 21x 3. Uniform Distribution 均匀分布 If the continuous random variable x with probability density function 46 1 for axb, f(x)ba0 elsewhere,Called X in the (a, b) obey uniform distribution f(x) 1 ba0 a b x Figure 4.2.1 The uniform probability density in the interval (a, b) 4. Exponential Distribution 指数分布 Many random variables, such as the life of automotive parts, life of animals, time period between two calls arrives to an office, having a distribution called exponential distribution. Definition 4.5.1 A continuous variable X has an exponential distribution with parameter (0), if its density function is given by x1e for x0 (4.5.1) f(x)0 for x0e01xdx1. Theorem 4.5.1 The mean and variance of a continuous random variable X having exponential distribution with parameter is given by E(X), D(X)2. 47 概率论与数理统计 连续型随机变量 连续型随机变量出现在我们交易量是衡量一个连续的规模。例如，我们测量汽车的速度时，一个人血液中的酒精含量，新合金的抗拉强度。 连续型随机变量 1.定义 定义4.1.1函数f(x)定义在(, )上，且满足下面两个条件 (i)对于任意的xR，都有f(x)0 ; (ii)f(x)在(, )上可积且f(x)dx1. 那么f(x)就叫概率密度函数. 定义4.1.2设f（x）是一个概率密度函数。如果x是一个具有分布函数的随机变量且 xF(x)P(Xx)f(t)dt (4.1.1) 那么x称为一个具有连续随机变量的密度函数f（x）。在这种情况下 x2P(x1Xx2)f(t)dt. (4.1.2) x12.范例 范例4.1.2发现K，以下可以作为一个连续随机变量的概率密度： f(x)k (x) 21x 3.Uniform Distribution 均匀分布 若连续型随机变量x具有概率密度函数 1 axb, f(x)ba0 其它则称x在（a,b）上服从均匀分布 48 4.Exponential Distribution 指数分布 许多随机变量，如汽车部件的生活，生活的动物，在两个通话时间到达办公室，具有分布为指数分布。 定义 4.5.1一个连续变量X有一个参数的指数分布，其密度函数为 x1e for x0 f(x)0 for x0e01xdx1. 定理 4.5.1连续型随机变量X具有指数分布的均值和方差由下式给出 E(X), D(X)2 49