A probabilistic inequality

forumfann

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

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

bpet

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?