[2017年整理]4.2 随机变量的函数的数学期望.ppt

二、二维随机变量函数的数学期望 三、数学期望的性质 * Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 一、一维随机变量函数的数学期望 例4.2 设随机变量 X 的分布律为 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 解 则有 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. (1)若X是离散型随机变量，且 X 的概率分布为 (2)若X是连续型随机变量，且其概率密度为 f(x)， 则 则 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 解 X -2 -1 0 0.1 P 1 0.2 0.3 0.4 例4.3 设随机变量 X 的概率分布如下： Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 解 例4.4 设随机变量 X 的概率密度为拉普拉斯分布 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 解 例4.5 游客乘电梯从底层到电视塔顶层观光，电梯于每个整点的第5分钟、25分钟和 55分钟从底层起行．假设有一游客在早上8点的第X分钟到达底层等候电梯，且X在[0,60]上均匀分布，求该游客等候时间的数学期望． 以Y 表示游客的等候时间，则 故 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. (1) 若(X,Y)是离散型随机变量，且其联合分布律为 则 (2) 若(X,Y)是连续型随机变量，联合概率密度为f(x,y)，则 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 解 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 1 x y 解 例4.7 设随机变量(X,Y)的联合概率密度为 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 1 x y 解 例4.7 设随机变量(X,Y)的联合概率密度为 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 性质 E(C)=C，其中C是常数。 性质 设X、Y独立，则 E(XY)=E(X)E(Y); 性质 若k是常数，则 E(kX)=kE(X); 性质 E(X1+X2) = E(X1)+E(X2); (诸Xi 独立时) 注意: E(XY)=E(X)E(Y)不一定能推出X,Y 独立 推广： Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copy