Choose a bounded strictly increasing function, for example $h(x)=\arctan x$. Independence and identical distribution imply $\operatorname{Var}(h(X)-h(Y))=2\operatorname{Var}(h(X))$. If $X=Y$ almost surely, the left side is zero. Therefore $h(X)$ is almost surely constant; injectivity of $h$ makes $X$ almost surely constant. Using a bounded function avoids any unassumed second moment of $X$.


Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 24J / a / Solution

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