设随机变量X Y和随机变量θ之间的关系为X=cosθ Y=sinθ 并设θ在0至2π范围内均匀分布 试说明X和Y不是统计独立

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设随机变量X,Y和随机变量θ之间的关系为X=cosθ,Y=sinθ,并设θ在0至2π范围内均匀分布,试说明X和Y不是统计独立的,但却是不相关的。

热搜:回归分析是分析一个随机变量和一个控制变量之间

答案解析

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  • 已知X=cosθ,Y=sinθ,且latex.action?latex=ZlxsZWZ0KCBcdGhldGEgIFxyaWdodCk9XGxlZnRce1xiZWdpbnthcnJheX17Y2N9XGZyYWN7MX17%0D%0AMlxwaX0sICAgJjBcbGVxIFx0aGV0YVxsZXEyXHBpIFxcMCwgICAgJiDlhbblroNcZW5ke2FycmF5%0D%0AfQ%3D%3D
       (1)要说明X和Y不是统计独立的,需证明P(X,Y)≠P(X)P(Y),即P(X≤a,Y≤b)≠P(X≤a)P(Y≤b),或P(cosθ≤a,sinθ≤b)≠P{cosθ≤a}P{sinθ≤b}。
       ①令a=b=0,则
       latex.action?latex=UFxsZWZ0KCBjb3NcdGhldGFcbGVxIDAgXHJpZ2h0KT1QXGxlZnQoIFxmcmFje1xwaX17Mn1cbGVx%0D%0AXHRoZXRhXGxlcSBcZnJhY3szfXsyfVxwaVxyaWdodCk9XGludF97XGZyYWN7XHBpfXsyfX1ee1xm%0D%0AcmFjezN9ezJ9XHBpfVxmcmFjezF9ezJccGl9ZFx0aGV0YT1cZnJhY3sxfXsyfQ%3D%3D
       latex.action?latex=UFxsZWZ0KCBzaW5cdGhldGFcbGVxIDAgXHJpZ2h0KT1QXGxlZnQoIFxwaVxsZXFcdGhldGFcbGVx%0D%0AIDJccGlccmlnaHQpPVxpbnRfe1xwaX1eezJccGl9XGZyYWN7MX17MlxwaX1kXHRoZXRhPVxmcmFj%0D%0AezF9ezJ9
       而
       latex.action?latex=UFxsZWZ0KCBjb3NcdGhldGFcbGVxLHNpblx0aGV0YVxsZXEgMCBccmlnaHQpPVBcbGVmdCggXHBp%0D%0AXGxlcVx0aGV0YVxsZXEgXGZyYWN7M317Mn1ccGlccmlnaHQpPVxpbnRfe1xwaX1ee1xmcmFjezN9%0D%0AezJ9XHBpfVxmcmFjezF9ezJccGl9ZFx0aGV0YT1cZnJhY3sxfXs0fT1QXGxlZnQoIGNvc1x0aGV0%0D%0AYVxsZXEgMCBccmlnaHQpUFxsZWZ0KCBzaW5cdGhldGFcbGVxIDAgXHJpZ2h0KQ%3D%3D
       ②但是,若令latex.action?latex=YT1iPVxmcmFje1xzcXJ0ezJ9fXsyfQ%3D%3D,则
       latex.action?latex=UFxsZWZ0KGNvc1x0aGV0YVxsZXEgXGZyYWN7XHNxcnR7Mn19ezJ9ICBccmlnaHQpPVBcbGVmdChc%0D%0AZnJhY3tccGl9ezR9XGxlcSBcdGhldGFcbGVxIFxmcmFjezd9ezR9IFxwaVxyaWdodCk9XGZyYWN7%0D%0AM317NH0%3D
       latex.action?latex=UFxsZWZ0KHNpblx0aGV0YVxsZXEgXGZyYWN7XHNxcnR7Mn19ezJ9ICBccmlnaHQpPVBcbGVmdCgw%0D%0AXGxlcSBcdGhldGEgXGxlcSBcZnJhY3tccGl9ezR9XHJpZ2h0KStQXGxlZnQoIFxmcmFjezN9ezR9%0D%0AXHBpXGxlcSBcdGhldGFcbGVxIDJccGkgXHJpZ2h0KT1cZnJhY3szfXs0fQ%3D%3D
       而
       latex.action?latex=UFxsZWZ0KGNvc1x0aGV0YVxsZXEgXGZyYWN7XHNxcnR7Mn19ezJ9LCBzaW5cdGhldGFcbGVxIFxm%0D%0AcmFje1xzcXJ0ezJ9fXsyfSBccmlnaHQpPVBcbGVmdFx7IFxmcmFjezN9ezR9XGxlcSBcdGhldGEg%0D%0AXGxlcSBcZnJhY3s3fXs0fVxwaSBccmlnaHRcfT1cZnJhY3sxfXsyfVxuZXEgUFxsZWZ0KCBjb3Nc%0D%0AdGhldGFcbGVxIFxmcmFje1xzcXJ0ezJ9fXsyfSBccmlnaHQpUFxsZWZ0KCBzaW5cdGhldGFcbGVx%0D%0AIFxmcmFje1xzcXJ0ezJ9fXsyfSBccmlnaHQp
       由此可见,X和Y不是统计独立的。
       (2)要说明X和Y不相关,需证明u11=0,因为
       latex.action?latex=RVxsZWZ0KFhccmlnaHQpPVxpbnRfey1caW5mdHkgfV57XGluZnR5IH14ZlxsZWZ0KCB4IFxyaWdo%0D%0AdClkeD1caW50X3swfV57MlxwaX1jb3NcdGhldGFmXGxlZnQoIFx0aGV0YSBccmlnaHQpZFx0aGV0%0D%0AYT0w
       latex.action?latex=RVxsZWZ0KFlccmlnaHQpPVxpbnRfezB9XnsyXHBpfXNpblx0aGV0YWZcbGVmdCggXHRoZXRhIFxy%0D%0AaWdodClkXHRoZXRhPTA%3D
       故
       latex.action?latex=dV97MTF9PUVcbGVmdFsgXGxlZnQoIFgtbV97WH0gXHJpZ2h0KVxsZWZ0KCBZLW1fe1l9IFxyaWdo%0D%0AdCkgXHJpZ2h0XT1FXGxlZnQoIFhZIFxyaWdodCk9RVxsZWZ0KCBjb3NcdGhldGFzaW4gXHRoZXRh%0D%0AXHJpZ2h0KT1cZnJhY3sxfXsyfUVcbGVmdCggc2luMiBcdGhldGFccmlnaHQpPTA%3D