[概率论与数理统计-第七章随机变量的数字特征.ppt

第七章 随机变量的数字特征 第一节 数学期望 第二节 方差和标准差 第三节 协方差和相关系数 第四节 切比雪夫不等式及大数律 第五节 中心极限定理 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 第一节 数学期望 一、离散型随机变量的数学期望 引例1 分赌本问题(产生背景) A, B两人赌技相同, 各出赌金100元, 并约定先胜三局者为胜, 取得全部200元。由于出现意外情况, 在A胜2局B胜1局时, 不得不终止赌博, 如果要分赌金, 该如何分配才算公平? Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. A 胜 2 局 B 胜 1 局 前三局: 后二局: A A A B B A B B A 胜 B 胜 分析 假设继续赌两局,则结果有以下四种情况: A A A B B A B B A胜B负 A胜B负 A胜B负 B胜A负 B胜A负 A胜B负 B胜A负 B胜A负 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 因此, A 能“期望”得到的数目应为 而B 能“期望”得到的数目, 则为 故有, 在赌技相同的情况下, A, B 最终获胜的 可能性大小之比为 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 因而A期望所得的赌金即为X的 “期望”值, 等于 X 的可能值与其概率之积的累加. 即为 若设随机变量 X 为:在 A 胜2局B 胜1局的前提 下, 继续赌下去 A 最终所得的赌金. 则X 所取可能值为: 其概率分别为: Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 设某射击手在同样的条 件下,瞄准靶子相继射击90次, (命中的环数是一个随机变量). 射中次数记录如下 引例2 射击问题 试问:该射手每次射击平均命中靶多少环? Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5