Anonymous
Anonymous asked in 科學數學 · 1 decade ago

如何求邊際密度函數

X和Y的聯合機率密度函數為

f(x,y)=c*(y^2-x^2)*e^-y

-y<=x<=y

0<y<無限大

1.求X和Y的邊際密度函數

麻煩各位大大幫我解個惑...>_<

Update:

對不起~我想請問一下,為什麼X的邊際密度函數是從[|x|~∞]呢@@"?

那E(X)又是多少>"

1 Answer

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  • 1 decade ago
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    1. 先求c

    ∫[0~∞]∫[-y~y] f(x,y) dx dy = 1

    => ∫[0~∞] ∫[-y~y] c (y^2-x^2) e^(-y) dx dy =1

    => ∫[0~∞] (4c/3) y^3 e^(-y) dy = 1

    => (4c/3)( -y^3 - 3y^2 - 6y- 6)e^(-y) 代y=0~∞ 得 8c=1 => c= 1/8

    2. 求Marginal pdf

    (x,y)之範圍: y>0, -y <= x <= y 即 y=|x|上方區域

    P_X(x)= ∫[|x|~∞] f(x,y) dy

    = ∫[|x|~∞] (1/8) (y^2 - x^2) e^(-y) dy (integration by parts)

    = (1/8)(x^2-y^2 - 2y-2) e^(-y) 代 y= |x|~∞

    = 1/4(|x|+1) e^(-|x|) , -∞<x<∞

    P_Y(y)= ∫[-y~y] f(x,y) dx

    = (1/8) ∫[-y~y] (y^2-x^2) e^(-y) dx

    = (1/6) y^3 e^(-y) , y>0

    2009-06-02 10:26:48 補充:

    因f(x,y)在 y>0, -y <= x <= y範圍內才不是0 (其他處為0)

    即 y=|x| 上方區域 f(x,y) 才有值(不是0 )

    y= |x| 上方圖形為V字型(頂點(0,0))上方區域, 故y的變化範圍= |x| ~∞

    E(x)=0 (左右對稱)

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