Re: [機統] joint pdf 的問題
※ 引述《wuwu944 (灣)》之銘言：
: A point is selected at random from the bounded region between two curves
: y=x^2-1 and y=1-X^2.Let X be the x-coordinate and Y be the y-corrdinate
: of the point selected.
: (a)find the joint pdf of X and Y
: (b)find marginal pdf of X and of Y
: (c)are X and Y independent?
: 謝謝
(a)令 f(x,y)=k,k為常數,R:the region between two curves
1 1-x^2
∫∫f(x,y)dydx = ∫ ∫ k dydx
R -1 x^2-1
1
=∫ k(2-2x^2)dx
-1
2 |1
=k(2x-—x^3)|
3 |-1
=(8/3)k =1
=>f(x,y)=k=3/8,|y|≦1-x^2
∞ 1-x^2
(b)f(x)=∫ f(x,y)dy =∫ (3/8)dy =(3/8)(2-2x^2) ,-1≦x≦1
-∞ x^2-1
∞ (1-y)^1/2
f(y)=∫ f(x,y)dx =∫ (3/8)dx ,y≧0
-∞ -(1-y)^1/2
(1+y)^1/2
∫ (3/8)dx ,y≦0
-(1+y)^1/2
=(3/4)(1-y)^1/2 , 0≦y≦1
(3/4)(1+y)^1/2 ,-1≦y≦0
=(3/4)(1-|y|)^1/2 ,|y|≦1
(c)f(x,y)≠f(x)f(y)
∴X and Y are not independent
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