Friday, 22 May 2015

probability - X and Y are independent and follow U(0,1). Show P(f(X)>Y)=int10f(x)dx





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=RR1[0,1](x)1[0,f(x))(y)dydx


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