Conditional expectation of a convex function.

Question

say I have a variable $X \sim P(x)$ distributed following some distribution which I do not know. I can estimate the sample mean and variance etc.

Suppose I transform as $Y = \exp(X) -1$.

My question is: Is there an analytical formula to connect $\mathbb{E}\big[X\;|\; X>X_0\big]$ to $\mathbb{E}\big[Y\;|\;Y>Y_0\big]$.

Thanks in advance.

Answer 1

Suppose that $X 1\{X>x_0\}$ has moments of all orders (e.g. $X$ is bounded). Then, assuming that $y_0=e^{x_0}-1$ and $\mathsf{E}[\exp(|X|)1\{X>x_0\}]<\infty$, $$ \mathsf{E}[Y\mid Y>y_0]=\frac{\mathsf{E}[Y1\{X>x_0\}]}{\mathsf{P}(X>x_0)}=\sum_{m\ge 1}\frac{\mathsf{E}[X^m 1\{X>x_0\}]}{m!\cdot\mathsf{P}(X>x_0)}=\sum_{m\ge 1}\frac{\mathsf{E}[X^m\mid X>x_0]}{m!}. $$

Answer 2

We have $$ \mathbb{E}\big[X\;|\; X>x_0\big] = \int_{x_0}^{+\infty} xP(x){\rm d}x $$ and $$ \mathbb{E}\big[Y\;|\;Y>y_0\big] = \int_{\ln(y_0+1)}^{+\infty} \left({\rm e}^x-1\right)P(x){\rm d}x. $$ At least, the conditional Jensen inequality holds: $$ \mathbb{E}\big[Y\;|\;Y>y_0\big]\le\exp\left(\mathbb{E}\big[X\;|\; X>x_0\big]\right)-1 $$ for $y_0 = {\rm e}^{x_0}-1$.