Markov's inequality over sum of two functions of a RV

Question

I'm wondering whether Markov's inequality can be applied over the following example, as I need an upper bound for the probability determined by:

$ P( f_1(X) + f_2(X) \geq \alpha ) $

Above, X is a random variable with pmf $p_X$, and functions $f_1$ and $f_2$ are deterministic and well defined. (though, we don't know them)

May I just apply Markov's inequality and do:

$ P( f_1(X) + f_2(X) \geq \alpha ) \leq \frac{E_X[f_1(X) + f_2(X) ]}{\alpha} $

My intuition is telling me this is possible. But I may be missing something I don't know.

Thank you all for your comments.

Answer 1

If you can guarantee $f_1(x)+f_2(x) \geq 0$ for all values $x$ that $X$ might attain, then yes. Define the random variable $Y=f_1(X)+f_2(X)$, which is non-negative, and apply Markov's inequality as usual: for $\alpha>0$, $P[Y>\alpha]\leq \mathbb{E}[Y]/\alpha$.

If you can't guarantee $f_1(x)+f_2(x) \geq 0$ then Markov's inequality won't hold.

Answer 2

The inequlaity is true if $f_1,f_2$ are measurable, $f_1(X)+f_2(X) \geq 0$ and $\alpha >0$. It is nothing but Markov's in equality applied to the random variable $f_1(X)+f_2(X)$.