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]．

2022-06-07

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]．

答案：

设[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]有联合概率密度为[tex=2.857x1.286]tj1rvgP4AHIdbrLux0kAEQ==[/tex]，则[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]的概率密度[tex=2.214x1.286]0oh9/Fj5WkMyNYlDmp1o0Q==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]的概率密度[tex=2.143x1.286]O2Tm6ZT1WQPSOA8P3c693w==[/tex]：[tex=10.0x2.429]DD6p67dNuoIoHGvXLciiMSFE1b7FTwd8DjFECL2yr01WVPJhpziohiACqtPPDTZ2qh7hszQRsFCY7/RJgUd0/g==[/tex]，[tex=10.0x2.429]w+B8dpIv0v1OhajW7HkNFegr2wpeRIpAOczCZoUfqfJrBFAzhtbBmVfob1hrOsRtjEQ8MtSYjApVcOHIukqGyQ==[/tex]．根据随机变量函数的计算公式，有[tex=18.643x2.429]Lf6OrLd25U7Z3QBuTvacPVUYoCFC0vDWkmIT8F8VwnU4BLG2YwnBESQ0qJWnir5V2om4sE5j/wSg9NFrcvmL+J5feE/ed9uRzEOOQ69XeNwemRI0r8fvoq63H1z10lbp[/tex][tex=23.429x2.429]6t4l9K4SHFb1evX8mwjMJnQA1UlYjW7t0rnPUzEtK4Xsfg1ZlVgARgTFoX9URJSf5it4ypUOT876ehZh+/FI++u6w72hoz2HYrHSPkXq69ui7XwzTNcQd0wr6G4jiSQjtEGLJk+e4NCo3RitUFGqsO545R+Q7GvUqi3MzKn9JpOJvfnStH/l2krabq0LTctXOLUGyjsD6N6s/Rb3/+q7Vt6GsEvn6t32195vLx/dokA=[/tex][tex=13.0x2.429]6t4l9K4SHFb1evX8mwjMJiy3xouLqYZmel9u3AlMSS5BVARbBFisOuWU1saMd+7n871rgEkafKuKEH0fxxdur+0wyVkb8D94yVFRndmsXbl//E+iIY+haF8yDOVXzt8ticMLTijACnu+mLsdvBeKLw==[/tex][tex=12.857x2.429]+MqSjsAYhX0PCyynny+5RE4fJa5AhFGhy8VpeB74ABMnZ2/OxpsDLE0fd/lwwta2a294VuKjPFO0sVhMWafAd2LNs7c69Lu6Vr6fkkvWLJ6GwQCSbBwQVSOLXQWbitjRIr3sr/U63tYUIFt5k6Bzig==[/tex][tex=14.929x2.429]6t4l9K4SHFb1evX8mwjMJnVfNW5G9JWd6EpXtYsSk7V5PoZ19ud2dnjkbqZQ7sVl+4E/NSDN5HnzHnql1WEaD7Z4eS3au6RpC+ncaP+sHoLUJTVyX0h95R2xeqk453hhpQdyR0ymlOE90CQ/Q1+HKw==[/tex][tex=5.429x1.286]WulMMDVOyVIMBbYdockJCw==[/tex]．假设[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]独立，即[tex=8.5x1.286]B5ivXD8CbpXxHSbT5muY1eHI9FCcuY/TJNp51wPYvO8=[/tex]，则[tex=14.929x2.429]I3sZ+nCtJuHRp2TI3VkoV9/gOtmLjQWPDkRVhBCFal0zDHV7JvaYmZi0HUZ775+6PveZwfhSqEvdsoBN5WeR0S28ROAEAKEfF3l3MbMPA3UtLUe2EXzUFUWYf9f4KA/i[/tex][tex=13.786x2.429]6t4l9K4SHFb1evX8mwjMJnQA1UlYjW7t0rnPUzEtK4Xsfg1ZlVgARgTFoX9URJSf2EG8MDX68LmSNAEKn7yxWX+s8isZLKfL8DlUHPIdqR/o+FHzYii5HlD5Z51nBZXfnU1VB4zHRfhTN86QHZWeTQ==[/tex][tex=13.929x2.429]6t4l9K4SHFb1evX8mwjMJnVfNW5G9JWd6EpXtYsSk7V5PoZ19ud2dnjkbqZQ7sVlug7P45B1j52R78wj7fFntoYqKouqv+MwQerHTGXhRkiwjHYzgPdtbcSFhLEjcxmWmMMalWMCx+a8I7jl6FPgAw==[/tex][tex=4.286x1.286]q3q63BxNDV7zQ8n16xFf0A==[/tex]．

举一反三

内容

0
设随机变量[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]与[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]相互独立，且[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]与[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]有相同的概率分布，其数学期望和方差存在，记[tex=4.929x1.286]coh7fE0sIReNY5IfTNUY2Q==[/tex]，[tex=5.0x1.286]w1pQ8Ky7lvfO3FrtoXXBqw==[/tex]，证明[tex=3.571x1.286]INBn7I2LD4mofTk9MYwDAWOnZiOE5Ty8TMG09ZPHuxo=[/tex]。
1
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？
2
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．
3
若[tex=0.929x1.286]uswT/CEcOIwMpCvTz/zeaA==[/tex]和[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]独立，证明[tex=19.143x1.286]NpVA38FZm94Nc/MNwvL8w8SoZ+pJnmEA8X0PISXKPg7Y8hNmBllpKcNorgYuDSrh[/tex]。
4
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．