Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Probability inequality for the sum of independent normal random variables

  1. Jul 24, 2007 #1
    Dear all,

    I wonder wheather there exsits a probability inequality for the sum of independent normal random variables ([itex]X_i[/itex] are i.i.d. normal random varianble with mean [itex]\mu[/itex] and variance [itex]\sigma^2[/itex]):
    [itex]
    P\left(\frac{1}{n}\sum_{i=1}^n X_i - \mu> \epsilon\right)\leq
    f(\epsilon, \sigma^2,n) \right).
    [/itex]

    We know that Bernstein inequality is for the sum of bounded random variables:
    [itex]
    P\left(\frac{1}{n}\sum_{i=1}^n X_i -\mu > \epsilon\right)\leq
    \exp\left(-\frac{n\epsilon^2}{2\sigma^2+ 2c\epsilon/3} \right).
    [/itex]

    I wonder whether there is some similar inequality for normal variables.

    Thanks!

    Phonic
     
  2. jcsd
  3. Jul 24, 2007 #2

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

    There is an exact equality; it follows from Σ X/n ~ N(μ, σ^2/n).
     
    Last edited: Jul 24, 2007
  4. Jul 26, 2007 #3
    Tanks for your reply. Then the problem is to bound the tail probability of this normal variable. I know one inequality is (R. D. Gordon, The Annals of Mathematical Statistics, 1941(12), pp 364-366)
    [itex]
    P(z \geq x) = \int_x^\infty \frac{1}{\sqrt{2\pi}}
    e^{-\frac{1}{2}z^2} dz \leq \frac{1}{x}
    \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}\mbox{\hspace{1cm}for } x>0,
    [/itex]
    where z is a standard normal variable.

    The problem of this inequality is that the function [itex]\frac{1}{x}
    e^{-\frac{1}{2}x^2} [/itex] is nor invertible (no analytical inverse function). Do you know some other bound for tail probability of a normal variable? Thanks a lot!

     
  5. Jul 26, 2007 #4

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

    Haven't you changed the upper bound function? Can the new function not have σ^2 or n as arguments? If it can, then you have an exact statement of the tail probability.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Probability inequality for the sum of independent normal random variables
Loading...