I am currently working through a proof of Markov's inequality which states that: $F(x)\leq \frac{E(x)}{x}$ and I am trying to understand how:
$\int^{\infty}_0xf(x)dx\geq \int^c_0xf(x)dx+c\int^\infty_0f(x)dx$
Where $F(x)$ and $f(x)$ are the cdf and pdf respectively.
I am confused about how the above inequality arises?
Note that for any $c\ge 0$, we have $$\int_0^\infty xf(x)\mathrm{d}x=\int_0^cxf(x)\mathrm{d}x+\int_c^\infty xf(x)\mathrm{d}x$$
Further, for $x\in [c,\infty)$, we have $x\ge c$, so $$\int_c^\infty xf(x)\mathrm{d}x\ge\int_c^\infty cf(x)\mathrm{d}x=c\int_c^\infty f(x)\mathrm{d}x$$
Putting these facts together, we have $$\int_0^\infty xf(x)\mathrm{d}x=\int_0^cxf(x)\mathrm{d}x+\int_c^\infty xf(x)\mathrm{d}x\ge\int_0^cxf(x)\mathrm{d}x+c\int_c^\infty f(x)\mathrm{d}x$$