在计算两个正态分布的AUC时，得不到相等的值

Question

我知道这有点棘手，但我希望你能为它做好准备。

给出两条正常曲线，我正在计算两条曲线平均值之间重叠的AUC。

首先，我计算两条曲线的截距的x值，如下所示：

gaussIsect = function(mu1,mu2,sd1,sd2){
    sd12 = sd1**2
    sd22 = sd2**2
    sqdi = sd12-sd22
    x1 = (mu2 * sd12 - sd2*( mu1*sd2 + sd1*sqrt( (mu1-mu2)**2 + 2*sqdi * log(sd1/sd2) ) )) / sqdi
    x2 = (mu2 * sd12 - sd2*( mu1*sd2 - sd1*sqrt( (mu1-mu2)**2 + 2*sqdi * log(sd1/sd2) ) )) / sqdi
    return(c(x1,x2))
}

然后我计算合并的AUC并找到x值，其中有一半：

gaussSplitOlap = function(mu1,mu2,sd1,sd2){
    # Make sure that mu1 < mu2
    if( mu1 > mu2 ){
        tmp = c(mu1,mu2)
        mu1 = tmp[2]
        mu2 = tmp[1]
        tmp = c(sd1,sd2)
        sd1 = tmp[2]
        sd2 = tmp[1]
    }
    # Get the intersect of the two normal curves
    isct = gaussIsect(mu1=mu1,mu2=mu2,sd1=sd1,sd2=sd2)
    # and make sure we have the intersect between the means
    isct = isct[which(mu1 < isct & isct < mu2)]
    # Calculate left curve AUC, i.e. from isct to inf
    a1 = 1-pnorm(isct,mean=mu1,sd=sd1)
    # Calculate right curve AUC, i.e. from -inf to isct
    a2 = pnorm(isct,mean=mu2,sd=sd2)
    # Calculate the total area
    A  = a1 + a2
    # Calculate the left curve x-value, which halfes the total area
    v1 = qnorm(1-A/2,mean=mu1,sd=sd1)
    # Calculate the right curve x-value, which halfes the total area
    v2 = qnorm(A/2,mean=mu2,sd=sd2)
    # Create and return list of results
    results = list(isct=isct,A=A,v1=v1,v2=v2)
    return(results)
}
test = gaussSplitOlap(mu1 = 10,sd1 = 0.9,mu2 = 12,sd2 = 0.6)
print(test)

如果我运行这个，那么测试就是

$isct
[1] 11.09291

$A
[1] 0.1775984

$v1
[1] 11.21337

$v2
[1] 11.19109

我无法弄明白，为什么v1！= v2？在我看来，它是从两个不同的参考点计算的相同值？