# Under what conditions we can add lower bounds of random variables?

Let $$X$$ and $$X_1,X_2, X_3$$ be non-negative random variables, and $$a_i, b_i$$ be also nonnegative scalars for $$i=1,2,3$$.

Suppose

$$\text{Prob}(X_1 \geq a_1) \leq b_1$$

$$\text{Prob}(X_2 \geq a_2) \leq b_2$$

$$\text{Prob}(X_3 \geq a_3) \leq b_3$$

and $$X \leq X_1+X_2+X_3$$.

Why is $$\text{Prob}(X \geq a_1 + a_2 + a_3) \leq b_1 + b_2 + b_3$$ true?

The reason I am asking is this paper Stochastic First-and Zeroth-order Methods for Nonconvex, where on pages 12 and 13, formulas (2.29-31) are the first three inequalities and (2.28) corresponds to the forth one. Using these formulas he shows (2.23).

If $$X \geq a_1+a_2+a_3$$ then $$X_1 \geq a_1$$ or $$X_2 \geq a_2$$ or $$X_3 \geq a_3$$. [Prove this by contradiction]. Probability of a union of events is less than or equal to the sum of the probabilities. Can you finish?
• @Saeed If $X_1 <a_1, X_2<a_2$ and $X_3 <a_3$ then $X \leq X_1+X_2+X_3 <a_1+a_2+a_3$. Apr 8, 2020 at 8:37