Let X and Y be two independent uniformly distributed r.v. on [0,1], and f is a continuous function from [0,1] to [0,1]. Show that P(f(X)>Y)=∫10f(x)dx.
I tried to prove it by change of variable but failed. I can only reach the step
P(f(X)>Y)=∫{(x,y):f(x)>y}I[0,1](x)I[0,1](y)dxdy
How can I proceed with the proof? Thanks for any help in advance.
Answer
∫(x,y):f(x)>y1[0,1](x)1[0,1](y)dxdy=∫R∫R1[0,1](x)1[0,f(x))(y)dydx
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