[EMAIL PROTECTED] (David) wrote in message news:<[EMAIL PROTECTED]>... > The original problem was indeed a trivariate normal. Can I take it > that covariances will always be zero if the variables are mutually > independent? > If two variables are independent then their covariance is zero. This is always true, even without normality. The converse is not true except under special conditions. One such condition is that the joint distribution is multivariate normal.
> Could the method here be extended to solve for a k-variate with k >3 ? > Rename the variables A,B1,...,Bk, and assume they are independent and normal. Let Xi = A - Bi, i = 1,...,k. Then Mean(Xi) = Mean(A) - Mean(Bi), Var(Xi) = Var(A) + Var(Bi), and Cov(Xi,Xj) = Var(A). The probability that A beats all of B1,...,Bk equals the probability that all of X1,...,Xk are positive, which is a k-dimensional integral that would have to be evaluated numerically. A different approach might be to work out the distribution of Bmax = max(B1,...,Bk) and evaluate Pr(A > Bmax) -- again, some fairly involved calculations. One of the features that makes this problem messy is that all the means and variances are different. > I am not sure how your representation (ie Integral_-u^Infinity > Integral_-v^Infinity f(x,y;w)dydx) translates to the 'usual' text book > representation for integrals. (bear in mind I a a novice here) > Here's an ascii attempt at the usual representation of the double integral (view it in a fixed-width font such as Courier): infinity infinity / / | |exp((-1/2)(x^2 + y^2 - 2wxy)/(1 - w^2)) | |--------------------------------------- dy dx | | 2 pi sqrt(1 - w^2) / / x = -u y = -v > Could these integrals be solved using numerical methods within > Microsoft Excel? > I don't use Excel, so I don't know, but I doubt it. However, even if it looks like Excel can do it, I'd be very wary because Excel has a reputation for using inaccurate algorithms. . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
