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A probabilistic inequality

  1. Dec 4, 2009 #1
    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\right)[/tex] for any [tex]\delta>0[/tex]?

    Any answer or suggestion will be highly appreciated!
    Last edited: Dec 4, 2009
  2. jcsd
  3. Dec 4, 2009 #2
    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.
  4. Dec 5, 2009 #3
    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] ?
    Last edited: Dec 5, 2009
  5. Dec 6, 2009 #4
    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?
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