What is an equation I can use for P((x+y)/2>n) in a multivarative normal distribution with a mean of 0, standard deviation of 1, and a correlational coefficient of .5?

Question

What is an equation I can use for P((x+y)/2>n) in a multivarative normal distribution with a mean of 0, standard deviation of 1, and a correlational coefficient of .5?

Answer 1

If you go to wikipedia (excellent source of mathematical formulae because it is non political) you will find that the formula for PDF of a multivariate normal distribution , restricted to the bivariate state (x,y) is as follows. (use link)

https://wikimedia.org/api/rest_v1/media/math/render/svg/bf91aebe806d84bb97ab95715e8e10e619387356

BUT! - to answer your question which is to predict the Probability P when (x+y)/2 > n you need the CDF. That is the rub!! There is no closed form for F(x,y). It can be calculated numerically (using iterations of the PDF) but there unfortunately is no F(x,y)

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