假设随机变量[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]的联合概率密度为[tex=17.571x3.143]EPaISH7F+7OFqeEao9lVbRHesk4tplA2VrcCvwQ3rO0t9Qq8Iw/niDFSpYDusNul2n6lMAa/nNo6fxngQQtlYClfavo3+nsShxM9BlAXlm07xYNG1+7omwt7s4WdO9vNijRJOmbFVFR9SeYuI5TFgQ==[/tex]．证明随机变量[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]不独立，但是[tex=1.286x1.286]ZIiW0MT/rNSURu/rNXyUxw==[/tex]和[tex=1.214x1.286]gnrbKJKP0x+Xz9YnDSiKgQ==[/tex]独立．
证：证[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]不独立．事实上，[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]的概率密度为[tex=9.571x2.286]DD6p67dNuoIoHGvXLciiMaolOxKLXIpLMy9nQh/e3LQib4rUA1oALJMO0lJI3bXCVdExA+vkM+ok8Jx3uuJbUg==[/tex][tex=7.786x3.143]39DN4CTrH1w5v3xsHjHTIicy02DCE788Mswj3Z+6yqIgRfQoS2T7WeSmRXipi0K7Rs4GyMYnyQ/WQdx7gBzrtflXpY5PccO+ji3Dhhf++UdJKsdIpOmMPqHVgA36smoE[/tex]；[tex=9.571x2.286]w+B8dpIv0v1OhajW7HkNFZVCAmKYG/tMPkEmvqh8awSjRy0Igyvl9djZkhYkThwpAX5DqX1temtwJxQPCymUvA==[/tex][tex=7.714x3.143]39DN4CTrH1w5v3xsHjHTIicy02DCE788Mswj3Z+6yqLlYQitiT5WkCsKQr/hUCnPuKZQkwkDYupVm10tAtLxU/g3Gpgyt8t52wz5gnwTl3f3YZw6HRGL5Ir61CWxJTvg[/tex]．由于[tex=8.5x1.286]UKP5fTDWON0hVvjW/SqgNxYPGMABCJpEyBe2sbfwXeNcn/1u2juUYNBDJG6wiS6x[/tex]，可见随机变量[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]不独立．(2)现在证明[tex=1.286x1.286]ZIiW0MT/rNSURu/rNXyUxw==[/tex]和[tex=1.214x1.286]gnrbKJKP0x+Xz9YnDSiKgQ==[/tex]独立．以[tex=2.357x1.286]DStX9aUQlkFNNagIrRNnXw==[/tex]，[tex=2.286x1.286]BdYGOjhEKXAvKu5esAslQA==[/tex]和[tex=3.0x1.286]5oF3fH+wIfZQrPGMBs+pmQ==[/tex]分别表示[tex=1.286x1.286]ZIiW0MT/rNSURu/rNXyUxw==[/tex]，[tex=1.214x1.286]gnrbKJKP0x+Xz9YnDSiKgQ==[/tex]和[tex=3.714x1.286]rRTEd1V/kF6YPRoDKfCekS+ebVBCd/mQKhCEIicsung=[/tex]的分布函数．显然，对于[tex=2.357x1.286]HiuPLo6AceUTMKkJXEjRDtRogWhBQcronuG2DqtdOsg=[/tex]，[tex=4.143x1.286]JPOCA1oC1A1FZ3wPU05FDw==[/tex]；对于[tex=2.357x1.286]mGWgTRWfwBrHxnj3eIqGjj+tJlm6ANIEEQWD+Fpm4fQ=[/tex]，[tex=4.143x1.286]CRyJ27AQcQJANFJz9yyAig==[/tex]；设[tex=4.071x1.286]TP6Xwsk2dd2Ql6CH6LqAgA==[/tex]，有[tex=8.857x1.286]Ps0wrQxyVn3voITBfIlTkCneM8trCrsoKen8QhkuyU+3GIwEDpVZ8DyHRCKI3H2S[/tex][tex=6.857x1.286]I11pzRKAIFiOkoca/35tFdjybZQgLVXoIbFl80UznD8kEbUAfICkfT2PK//yKUrh[/tex][tex=7.929x2.643]QhSn3nQm91RWQg5gFzFNUM5kK/kdm6AofHMo9XJ9qo6e4t2r5TifkfaswMhSQyn/IIA6b9TU9dZM4cTj305MbQ==[/tex]．[tex=12.286x3.643]xcJprc4S9bSa2ociAv2kD/jwCbp5P0Gb7AjFi3KSAfl6itXk62FNCgd18BwyRzS25vBvXE93C3TwXO+Yk6eUQQF73TVSSMYIMJBv8JZmZYQqR5Duk3ZirBaBJAenDuSIsahlzFOPG0H5FrtuWMzPJg==[/tex]；[tex=12.143x3.643]Slt3soq4B0fWsjJnjVqEnaq4VeOMalH/Nc8f5rTkaClHsfHcualGGBNeK0vd8EEsS4EAeDQQCG+6JNJuNYsnmWJopobrd/G18JM9mbuRJXbSzErefHM1ZYnhhmXb5cHiY4snYDy430XAJHkhdXll2Q==[/tex]；对于[tex=2.357x1.286]HiuPLo6AceUTMKkJXEjRDtRogWhBQcronuG2DqtdOsg=[/tex]，[tex=2.286x1.286]kd5QHEOmRC4ZvcbW9S/u+u9jwLE7gCUSaNuwKhEufhE=[/tex]，[tex=4.786x1.286]8hITF+90NOgi9KguBCKkSw==[/tex]；对于[tex=2.357x1.286]mGWgTRWfwBrHxnj3eIqGjj+tJlm6ANIEEQWD+Fpm4fQ=[/tex]或[tex=2.357x1.286]vWlyphRcu4jsIwuJh4IqunCqxd8q+ORDnfDRH/XPPtQ=[/tex]，[tex=4.786x1.286]3atyKY6MmO5vFoDq7UL7eQ==[/tex]；对于[tex=4.071x1.286]awTKfklzFH1mxgNaIGMS0pLnYQyVzRAeK1qInCG+vwc=[/tex]，[tex=2.286x1.286]kd5QHEOmRC4ZvcbW9S/u+u9jwLE7gCUSaNuwKhEufhE=[/tex]，有[tex=17.286x2.643]TChxD1GcYwxh4S4Ic5P4lPj8H/Fdi/q9d/NKKDPqvPu+1tw42X4MAUSjvgczipGAwzgq2Jfy9sjITi6j7jKk8piDdXNEtfGEE3/EzZkZuzr7wzeEMxyTc9aoa33xlngidVh98hoqJcuDhYOMZIg30Q==[/tex]；同理，对于[tex=4.071x1.286]2/BPLf7gTGFS59kisYHLr1u0stOkypZxf+8cNceFT/o=[/tex]，[tex=2.357x1.286]HiuPLo6AceUTMKkJXEjRDtRogWhBQcronuG2DqtdOsg=[/tex]，有[tex=5.714x1.286]5lxqzsNuoMov9YwySuXguHtvs9FZV7hlKPnryLm2g9Q=[/tex]；对于[tex=4.071x1.286]awTKfklzFH1mxgNaIGMS0pLnYQyVzRAeK1qInCG+vwc=[/tex]，[tex=4.071x1.286]2/BPLf7gTGFS59kisYHLr1u0stOkypZxf+8cNceFT/o=[/tex]，有[tex=12.929x1.286]lUtsMm/NpEv/Qdopxk8t5JpwzL1h8Snocc42a4GeK9vbNIv3LvEeo4du+icjyu+dGX59sZMWFCFj2+1lj4WP3pzieJ9orqxeWGKoTT8f59U=[/tex][tex=15.071x2.714]FlBqqsvhcKHs/Fz3lzriE1Kgrg5RkAloiFkfNHmCj34jAwUAyPymM0dL6dvtjxmN7hLTFRX4VKcYdML70jSHx/4bipzjEvfjemPlQl+CsoyFxiF1vlmHScQqQNdqbkmAaj+gYvNak+vzWDjDKcELTA==[/tex]．于是[tex=1.286x1.286]ZIiW0MT/rNSURu/rNXyUxw==[/tex]和[tex=1.214x1.286]gnrbKJKP0x+Xz9YnDSiKgQ==[/tex]的联合分布函数为[tex=18.0x5.929]Ze2PMtFTJCSB/yFpS+6VrfTY87SzouSMiZRfTfFZFp5BtiIaAtofrjzxaHWy3qhkhiAjlHFwIyqAr3sm4IMXxn57a5mGphFPK4jICOBKqd+lxV8N1ifHhtaOFBulrt51QRCiIQS4ywdM4RVGmSyTWeEk70p7HOz79xT5erO7cGbl7sgqafbew4JofpfzuTDTeVvFeaY/TTqhJP5fwC7TC47egqSVRa1aiPsX4lygFO45rzBrmO9m+3DmCVtgOmjznqL+rb9sxs8gPgXk8ILqPVCN02L6de1VNpOS1g4ZuaH7mHyhhG2T1gLz+amdnbxf[/tex]．由此可见，[tex=9.0x1.286]CrJbPtpjgL9MSRq7MYoZR6Sz014s4jAofJMWUpQY+9Y=[/tex]，因此随机变量[tex=1.286x1.286]ZIiW0MT/rNSURu/rNXyUxw==[/tex]和[tex=1.214x1.286]gnrbKJKP0x+Xz9YnDSiKgQ==[/tex]独立．
举一反三
- 设随机变量[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]都服从[tex=2.143x1.286]dboSCjP3Fn5+xkkJFCNE+A==[/tex]分布，证明： “[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]不相关”与“[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]独立”等价．
- 设[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]是两个相互独立的随机变量，[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]在[tex=2.929x1.286]kvrkODQf0L3CKREOEdSkuA==[/tex]上服从均匀分布，[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]的概率密度为[tex=10.571x2.429]DRJq+C1mHjswrEZ8FtvX7HNGAPrBLJ6gzRGG2ilTN7MM55jZEydQmT0AUl0Qb5hAT5k9ols3J/KpgflWFdX4TQ==[/tex]，求：（1）[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]的联合概率密度；（2）[tex=4.714x1.286]dbgFLPFxgdKKXnbc/gnthjs3iie6rgn/UEwrXH27vHI=[/tex] .
- 设[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]，[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]相互独立，且[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]服从数学期望为150 , 方差为9的正态分布，[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]服从数学期望为100，方差为16的正态分布，求[tex=7.5x1.357]JgfvMEzlJt4TFydcPQ2gaw==[/tex]，[tex=10.286x1.357]/kMGdCxDBv+iw/Cr+hQeUnIAq7x/u//czEtqpBiPB/0=[/tex]。
- 若随机向量[tex=2.786x1.286]d8ZGztHRaPoTHI8v2JwIGQ==[/tex]服从二维正态分布，则①[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]，[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]一定相互独立；②若[tex=3.5x1.286]sKaD0gq7ZfmqhDuxwY0565jK5tQQMeY1a44eA15r+0I=[/tex]，则[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]，[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]一定相互独立；③[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]都服从一维正态分布；④若[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]，[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]相互独立， 则[tex=6.143x1.286]1FUpcitV2qNzFaSobaOhNfKUbfF8QOwkW6yD2rc0W2g=[/tex]，几种说法中正确的是 A: ①②③④ B: ②③④ C: ①③ D: ①②④
- 设随机变量[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]的数学期望[tex=1.714x1.286]tnqXjXNHESmtAydX2nd1FQ==[/tex]和[tex=1.571x1.286]9HHQOQ6kFW8m23SI56qi0g==[/tex]存在，证明：假如[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]独立，则[tex=7.0x1.286]Fsc4c/MsrMbL1SEpyKHrDmKSWwNmUF4ydiRy0R1FUw0=[/tex]．
内容
- 0
设[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]分别表示100次独立重复试验成功次数和失败的次数，证明：随机变量[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]的标准差的最大值等于5，相关系数等于[tex=1.214x1.286]WDa3CFFbujv+acHNTSW8sQ==[/tex]．
- 1
设随机变量[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]的方差存在，证明：[tex=10.143x1.286]HG2ihwjcXTdzCTS/bC0QJsaC65j3BHkkW1/8B8OIxFg=[/tex]是[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]不相关的充分和必要条件．
- 2
设[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]是只有两个可能值的离散型随机变量，[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]是连续型随机变量，而且[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]相互独立，证明随机变量[tex=4.929x1.286]bstb6Acm/GnARrPc8f1uPw==[/tex]是连续型随机变量．
- 3
设二维随机变量[tex=2.786x1.286]vzGOG+JNlRurOKCm31T4Kw==[/tex]在圆域[tex=5.357x1.286]oOYTzm/NiJqJo4OjC55er1L5z17HiYuK5dHQrlDB2IM=[/tex]上服从均匀分布，（1）求[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]的相关系数[tex=0.571x1.286]mGHbklYlBVNXKEGAelwITA==[/tex]；（2）问[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]，[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]是否独立？
- 4
设随机变量 [tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex] 与 [tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex] 相互独立， [tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex] 服从正态分布 [tex=3.929x1.286]N5dq4BwkTdWMAb0OmXWoEaQHcjMspfC0l4+u6bRl6uAvEVUQUcSxPV1hL5aXeKrf[/tex], [tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex] 服从均匀分布 [tex=3.857x1.286]oINv2OUrkfWf54e8Ht2lD1iv2R1pi2JiMcP1OIfioeI=[/tex] , 求 [tex=4.929x1.286]bstb6Acm/GnARrPc8f1uPw==[/tex] 的密度函数.