TITLE: Integral inequation
QUESTION [5 upvotes]: In my statistics book Chebyshev's inequality is proven. In several steps this inequality is used:
$$ \int_a^{+\infty} \phi(x) f_X(x)dx \quad \geq \quad \phi(a) \int_a^{+\infty} f_X(x)dx $$
and also:
$$ \int_{-\infty}^{-a} \phi(x) f_X(x)dx \quad \geq \quad \phi(-a) \int_{-\infty}^{-a} f_X(x)dx $$
Here is $a \geq 0$, $\phi:\mathbb{R}\to\mathbb{R}^+$ a positive function, and $f_X$ a pdf.
Why this is valid?

REPLY [3 votes]: Since $\phi$ is increasing (this is important here) and $f_X$ is positive, we have for $a>0$ and $x\ge a$, $\phi(x)\ge\phi(a)$. Therefore,
$$
\begin{align}
\int_a^\infty\phi(x)f_X(x)\,\mathrm{d}x
&\ge\int_a^\infty\phi(a)f_X(x)\,\mathrm{d}x\\
&=\phi(a)\int_a^\infty f_X(x)\,\mathrm{d}x\\
\end{align}
$$
The other inequality is simply a change of variables.