Markov's Inequality

If $x$ takes only Nonnegative values, then

$$P(x \geq a) \leq {\left\langle{x}\right\rangle{}\over a}.$$

To prove the theorem, write

$$\langle x\rangle = \int^\infty_0 xf(x)\,dx = \int^a_0 xf(x)\,dx+\int^\infty_a xf(x)\,dx.$$

Since $P(x)$ is a probability density, it must be $\geq 0$. We have stipulated that $x \geq 0$, so

$$\left\langle{x}\right\rangle{} = \int_0^a xf(x)\,dx+\int_a^\infty xf(x)\,dx$$

$$\geq \int^\infty_a xf(x)\,dx \geq \int^\infty_a af(x)\,dx$$

$$= a\int^\infty_a f(x)\,dx = aP(x \geq a),$$

Q. E. D.