Assume that $d$ is a random variable with pdf $f_d(x)$, is the following true?
$$E\left[\frac{1}{d^2}\bigg|d>1\right]\leq E\left[\frac{1}{d^2}\bigg|d>1 \cap d<2\right]$$
My answer is yes, since by additional condition on the right side of above equation we are deleting large $d$ area therefore, removing small $\frac{1}{d^2}$ area. Thus, the average should increase. Is that right? How can I see a mathematical solution?
Begin with: the definition of conditional expectation when given an event, and the linearity of expectation.
$$\begin{align}\mathsf E(d^{-2}\mid 1<d<2) & = \dfrac{\mathsf E(d^{-2}\mathbf 1_{1<d<2})}{\mathsf P(1<d<2)} \\[1ex] & =\dfrac{\mathsf E(d^{-2}\mathbf 1_{1<d}-d^2\mathbf 1_{d\geqslant 2})}{\mathsf P(1<d<2)} \\[1ex] & = \dfrac{\mathsf P(d>1)\mathsf E(d^{-2}\mid d>1)-\mathsf P(d\geqslant 2)\mathsf E(d^{-2}\mid d\geqslant 2)}{\mathsf P(1<d<2)}\\[1ex] & = \end{align}$$
Take it from there.