Inequality on probability of the sum of random variables

Question

For Independent random variables $X_{i}$ can we write down the following inequality? $$\Pr \left\{ {{X_1} + {X_2} + ... + {X_n} \le k} \right\} \le \Pr \left\{ {{X_1} \le k} \right\}\Pr \left\{ {{X_2} \le k} \right\}...\Pr \left\{ {{X_n} \le k} \right\}$$

Answer 1

It is true if your random variables are nonnegative: $$ X_1+X_2+\cdots+X_n\leq k\implies X_1\leq k\;\&\;X_2\leq k\;\&\;\ldots\;\&\;X_n\leq k. $$ So $$ \Pr(X_1+X_2+\cdots+X_n\leq k)\leq\Pr(X_1\leq k\;\&\;X_2\leq k\;\&\;\ldots\;\&\;X_n\leq k)=\prod_{i=1}^n\Pr(X_i\leq k). $$ The equality above uses independence.

Answer 2

The proposition you provide is correct if each of the $X_i$ can only take on non-negative values. (I'm sure there are other more complicated conditions that would also make it correct, but for general distributions, the inequality does not hold.)

A counterexample with $n=2$: Let $X_1$ and $X_2$ each be uniformly disributed on $-3,-2$, and choose $k = -4.5$. The right hand side is zero, the left hand side is not.