Expectation of the sum of two random variables whose probabilities are individualy bounded

Question

I encountered the following problem:

I have two random variables $A$ and $B$. I know only that $$P(|A|>\epsilon)\leq a(\epsilon)$$ and $$P(|B|>\epsilon)\leq b(\epsilon)$$ where $a$ and $b$ are known functions dependending on $\epsilon$.

Can I say something about the expectation of $E[A+B]$? Also an inequality involving $E[A+B]$ would be helpful!

You can say that $E[|A+B|] \leq E[|A|+|B|]=E[|A|]+E[|B|]=\int a(x)dx+\int b(x)dx.$ See this for the last equality. — Austin80, Mar 31 '20 at 13:39
Sorry, above the last equality should in fact be an inequality. — Austin80, Mar 31 '20 at 15:55

score 1 · Accepted Answer · answered Mar 31 '20 at 13:36

1

Note that $|A+B| > 2 \epsilon$ implies $|A| > \epsilon$ or $|B| > \epsilon$, so $P(|A+B| > 2 \epsilon) \le a(\epsilon) + b(\epsilon)$. This in turn implies $$E|A+B| = \int_0^\infty P(|A+B| > x) \; dx \le \int_0^\infty (a(x/2) + b(x/2))\; dx$$

answered Mar 31 '20 at 13:36

Robert Israel

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Can it be generalized to find $E|A+B|^p$, $p>1$? – volond Mar 31 '20 at 14:05

Expectation of the sum of two random variables whose probabilities are individualy bounded

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