Does This Probabilistic Inequality Hold for IID Random Variables?

[forumfann]

Suppose x_1,x_2,x_3,x_4 are non-negative Independent and identically-distributed random variables, is it true that <br /> P\left(x_{1}+x_{2}+x_{3}+x_{4}&lt;2\delta\right)\leq2P\left(x_{1}&lt;\delta\right) for any \delta&gt;0?

Any answer or suggestion will be highly appreciated!

 

[bpet]

This might well hold without the independence assumption. Use x1+x2+x3+x4>=x1+x2 then consider the cases x1<d and x1>=d separately.

 

[forumfann]

Thanks. But then is it true that P\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}&lt;3\delta\right)\leq2P\left(x_{1}&lt;\delta\right) for any \delta&gt;0 ?

 

[bpet]

Thanks. But then is it true that P\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}&lt;3\delta\right)\leq2P\left(x_{1}&lt;\delta\right) for any \delta&gt;0 ?

This is not easy. Change the 2 to 3 and it is certainly true (using same method as before). What if the variables are Bernoulli, does the inequality hold?