Theorem: Let XXX is a random variable with CDFCDFCDF FX(x)F_X(x)FX(x), then Y=FX(X)∼U[0,1]Y=F_X(X)\sim U[0,1]Y=FX(X)∼U[0,1]
proof: CDFCDFCDF of YYY, SY=[0,1]S_Y=[0,1]SY=[0,1] (∵\because∵ 0≤FX(x)≤10\le F_X(x)\le 10≤FX(x)≤1)
for 0≤y≤10\le y\le10≤y≤1, FY(y)=P(Y≤y)F_Y(y)=P(Y\le y)FY(y)=P(Y≤y)
=P(FX(X)≤y)=P(F_X(X)\le y)=P(FX(X)≤y)