I have a normal distribution of a variable y~N(f(x),bc^2x), where x is the IDV and b and c are some estimated parameters. So, the SD of y is given by bc^x.

Now I have a transformation of y:

z=g(y,n)=e^y(n+1)/(1+e^y)/n, where n is another IDV.

In this case, what is the best way to estimate the SD of z? Or, what is the "?" in z~(e^f(x)(n+1)/(1+e^f(x))/n,?) ?

I think it would be the fastest if I can have a closed form equation for the SD of z but I don't know how to do the transformation.

Meanwhile, I can think of the following alternatives:

1. Monte Carlo Simulations, but it is time-consuming, 2. I am wondering if 0.5[g(f(x)+bc^x,n)-g(f(x)-bc^x,n)] can be regarded as a reasonable estimator for SD of z

Any help is appreciated.