[問題] f_X,Y與F_Z的關係

[buttermilk (脫脂牛奶)]

f_X,Y(x,y) = 1, 0 < x < 1, 0 < y < 1

           = 0, elsewhere

Let Z = X + Y.

F_Z(z) = P(Z≦z) = P(X+Y≦z).


Then F_Z(z) = 0                            z < 0
               z z-x
            = ∫∫dydx = (z^2)/2           0≦z<1
               0 0
                 1 1
            = 1-∫∫dydx = 1-[(2-z)^2]/2   1≦z<2
                z-1z-x

            = 1                            2≦z

請問f_X,Y(x,y)是怎麼變成F_Z(z)的？

我看不太懂F_Z(z)等式後面那些積分的上下標範圍。

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