Conditional expectation $E[X^2|X>1]$

Question

Given that $X$ is a exponential random variable. I want to calculate $E[X^2|X>1]$. I think it would be $E[(X+1)^2]$, but do not have any convincing explanation.

Answer 1

Memorylessness. The conditional distribution of $X$, given that $X\gt 1$, is the same as the distribution of the random variable $1+Y$, where $Y$ has exponential distribution with the same parameter as $X$. Informally, the additional waiting time, given that one has waited for time $1$, has the same distribution as $X$.