Using the formula \expec(Z)=∫∞0\prob(Z>x)dx that's valid for any nonnegative r.v.\ Z, we have that \begin{eqnarray*} \sum_{m=1}^\infty \frac{\var(Y_m)}{m^2} &\le& \sum_{m=1}^\inft...Using the formula \expec(Z)=∫∞0\prob(Z>x)dx that's valid for any nonnegative r.v.\ Z, we have that \begin{eqnarray*} \sum_{m=1}^\infty \frac{\var(Y_m)}{m^2} &\le& \sum_{m=1}^\infty \frac{1}{m^2} \expec(Y_m^2) =\sum_{m=1}^\infty \frac{1}{m^2}\expec\left(X_m^2 \ind_{\{X_m\le m\}}\right) \\ &=& \sum_{m=1}^\infty \frac{1}{m^2} \int_0^\infty \prob\left(X_m^2 \ind_{\{X_m\le m\}}>t\right)dt \\ &=& \sum_{m=1}^\infty \frac{1}{m^2} \int_0^\infty \prob\left(\sqrt{t} \le X_m \le m \rig…
