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:<math>H(z)=\int_{-\infty}^\infty F(z-t)g(t) dt = \int_{-\infty}^\infty G(t)f(z-t) dt</math>
If we start with random variables <math>X</math> and <math>Y</math>, related by <math>Z = X + Y</math>, and without knowledge of these random variables being independent, then:
:<math>f_Z(z) = \int \limits_{-\infty}^{\infty} f_{XY}(x, z-x)~dx</math>
However, if <math>X</math> and <math>Y</math> are independent, then:
:<math>f_{XY}(x,y) = f_X(x) f_Y(y)</math>
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