inequalities in probability

hedipaldi

New member
Joined
Aug 26, 2015
Messages
3
Hi,
I need help solving the attached probleb.
Thank's in advance
 

Attachments

  • 001.jpg
    001.jpg
    172.6 KB · Views: 4
I need help solving the attached problem.
I think the exercise is as follows:



X1, X2 are random variables with zero expectancies and standard deviations \(\displaystyle \sigma_1,\, \sigma_2\) respectively. The correlation coefficient of X 1 and X2 is \(\displaystyle \rho\).

Prove that, for k1, k2 > 0, we have:


. . . . .\(\displaystyle \displaystyle P\left(\bigcap_{i=1}^2\, \left|\dfrac{X_i}{\sigma_i}\right|\, <\, k_i \right)\, \geq\, 1\, -\, \dfrac{k_1^2\, +\, k_2^2\, +\, \sqrt{\strut \left(k_1^2\, +\, k_2^2\right)^2\, -\, 4 k_1^2 k_2^2 \,}}{2(k_1 k_2)^2}\)



Is this correct?

When you reply (with corrections or confirmation), please include a clear listing of your thoughts and efforts so far. Thank you! ;)
 
Reply to stapel

I don't have the probability distributions of the variables,so i tried using Markov and Chebyshev's inequalities.I also tried to use bounds on a quadratic form to find an upper bound outside the given rectangle.I t can be assumed that the expectancies are zero.
I really need a clue or some help.
 
Top