# Prove conditional expectation $E[X\mid X>x]$ is the unconditional expectation $E_{P^*}[X]$ under a probability measure $P^*$

Prove that the conditional expectation $$\mathbb E[X\mid X>x]$$ (here x is fixed, say x=10) is the unconditional expectation $$\mathbb E_{\mathbb P^*}[X]$$ under a probability measure $$\mathbb P^*$$. Derive a formula for $$\mathbb P^*$$ in terms of the original probability measure $$\mathbb P$$.

I can only think of taking double expectation on the conditional expectation to remove the condition, but I'm not sure if that's the right step to take. Any suggestion is appreciated.

• This is an homework question, isn't it? So please use the self-study tag. Commented Dec 13, 2014 at 7:24

$$\mathbb{P^*}$$ is defined as follows: $$\mathbb{P^*}(A) := \mathbb{P}(A\mid X > x) = \frac{\mathbb{P}(A\cap \{X > x\})}{\mathbb{P}(X > x)}, ~~~\forall A \in \mathcal{F}.$$

Of course, this definition is only valid when $$\mathbb{P}(X > x) > 0$$. It can be checked $$\mathbb{P^*}$$ defined in this way is indeed a probability measure on $$(\Omega, \mathcal{F})$$ and satisfies $$\mathbb{E}(X\mid X > x) = \mathbb{E}_{\mathbb{P^*}}(X)$$.