Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 2 / 11F / a / Solution

Write $\mu =\mathbb{E}[X]$ and $A=\{X>\theta \mu \}$. The finite second moment implies a finite first moment, and nonnegativity with $\mathbb{E}[X^2]>0$ implies $\mu >0$. On $A^c$, one has $0\le X\le \theta \mu$. ThereforeSubtract this from $\mathbb{E}[X]=\mu$ to obtain the truncated first moment bound:The indicator function uses the strict event specified in the question; any mass at the threshold belongs to $A^c$.