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 $$P\left(x_{1}+x_{2}+x_{3}+x_{4}<2\delta\right)\leq2P\left(x_{1}<\delta\right)$$ for any $$\delta>0$$?

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 $$P\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}<3\delta\right)\leq2P\left(x_{1}<\delta\right)$$ for any $$\delta>0$$ ?

 

[bpet]

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

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?