1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finding the P(2x1>x2) for a bivariate normal distribution

  1. Feb 7, 2012 #1
    1. The problem statement, all variables and given/known data

    Given a bivariate normal distribution with E(x1)=4 and E(x2) = 6 and Var(X) = [3 2.5]
    [2.5 7]
    Find P(2*x1>x2)

    2. Relevant equations

    The cdf of this bivariate normal distribution is given by:

    f(x1,x2)=1/(2*pi*var(x1)*var(x2)*sqrt(1-rho^2)) * e^(-0.5*(z/(2*1-rho^2)))

    where var(x1) = 3, var(x2) = 7, E(x1)=4 and E(x2) = 6, and rho = 2.5/(sqrt(3)*sqrt(7))

    and z = ((x1-E(x1))^2)/var(x1) + ((x2-E(x2))^2)/var(x2) - (2*rho*(x1-E(x1))*(x2-E(x2)))/sqrt(var(x1)*var(x2))

    3. The attempt at a solution

    The approximate cdf of this bivariate normal distribution is given (in terms of x and y) by:

    .041440417*%e^(-0.71186*((((x-4)^2)/3)+(((y-6)^2)/7)-(1.09109*(x-4)*(y-6))/4.582576))

    Taking the integral from (-infinity to 2*x) dy and then from (-infinity to + infinity) dx should do the trick, but I have been unable to do so even using approximations.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 7, 2012 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Can you figure out the distribution of the single random variable Y = 2*X1 - X2?

    RGV
     
  4. Feb 7, 2012 #3
    I can't, not sure if I am missing something or just being thick.
    Working only on z=((x1-4)^2)/3 + ((x2-6)^2)/7 - 5*(x1-4)(x2-6)/21

    Distributing I get z=(7x1^2+3x2^2-26x1-16x2-5x1x2+100)/21

    My best attempt has z = (2y^2-4y-x1^2+x2^2+3x1x2-20x2-18x1+100)/21
     
  5. Feb 7, 2012 #4
    Figured it out, thanks for the idea, it was extremely helpful :-)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Finding the P(2x1>x2) for a bivariate normal distribution
  1. Bivariate normal (Replies: 9)

Loading...