Is second moment of exponential also Memory Less?

Question

I know that the exponential is memory less and that means that: $$ E[X\mid X>1]=1+E[X] $$ Now, does the memory less property also hold for the second moment? Specifically, is the following true? $$ E[X^2\mid X\geq1]=1+E[X^2] $$

Answer 1

Actually it should say $$ \operatorname{E}(X^2\mid X\ge 1) = \operatorname{E}((1+X)^2). $$ The conditional distribution of $X-a$ given that $X\ge a$ is the same as the marginal (i.e. unconditional) distribution of $X$. Hence the conditional distribution of $X$ given that $X\ge a$ is the same as the marginal distribution of $a+X$.