Show for any two random variables X and Y, and for any constant c P{X+Y>c} ≤ P{X>c/2} +P {Y>c/2} Show that for any two random variables $X$ and $Y$, and for any constant $c\in\mathbb R$
$$P\{X+Y>c\} ≤ P\{X>\frac c2\} +P \{Y>\frac c2\}$$
Attempt:
Set $Z = X+Y$ hence,
$$P\{Z>c\} ≤ P\{Z-Y>\frac c2\} +P \{Z-X>\frac c2\}$$
$$P\{Z>c\} \leq P\{Z>\frac c2 + Y\} +P \{Z>\frac c2 + X\}$$
I'm not really sure how to show this or what the ending solution should even look like. 
 A: Note that
$$ P(X > c/2 \text{ or } Y > c/2) = P(X > c/2) + P(Y > c/2) - P(X > c/2 \text{ and } Y > c/2) \leq P(X > c/2) + P(Y > c/2).$$
Now if $X + Y >c$, then $X > c/2$ or $Y > c/2$ because if both were smaller than $c/2$, their sum couldn't be bigger than $c$. So $P(X + Y > c) \leq P(X > c/2 \text{ or } Y > c/2)$. Putting this together gives you
$$ P(X + Y > c) \leq P(X > c/2 \text{ or } Y > c/2) \leq P(X > c/2) + P(Y > c/2) $$
A: P{X + Y > c} = P{x > c/2 and y > c/2} + P{all other events that make X + Y > c} <= P{x > c/2 and y > c/2} = P{x > c/2} + P{y > c/2} - P{x > c/2 or y > c/2} <= P{X > c/2} + P{Y > c/2}
using the inclusion/exclusion formula and nonnegativity of probability
A: Hints:


*

*Draw a diagram of the plane with coordinate axes $x$ and $y$, and mark on it the region  in which the random point $(X,Y)$ must lie in order to satisfy the
condition that $X+Y > c$.  

*Next, mark on the same diagram the regions corresponding to the events
$\left\{X > \frac{c}{2}\right\}$ and $\left\{Y > \frac{c}{2}\right\}$ and verify that
$$\{X+Y > c\} \subset \left\{X > \frac{c}{2}\right\} \cup \left\{Y > \frac{c}{2}
\right\}.$$

*Then use $A \subset (C\cup D) \Rightarrow P(A) \leq P(C\cup D)$ and $P(C \cup D) \leq P(C)+P(D)$ to complete the proof.
A: Show the straightforward inclusion
$$
\left\{X\leq \frac c2\right\}\cap\left\{Y\leq \frac c2\right\}\subseteq \{X+Y\leq c\}
$$
and deduce, by looking at complements, that
$$
\{X+Y>c\}\subseteq\left\{X>\frac c2\right\}\cup\left\{Y>\frac c2\right\}.
$$
Then the inequality follows from basic properties of probability measures.
