I have a pair of 2 simple normal distributions for noise and signal , specified by $\mu1,\sigma1$ and $\mu2,\sigma2$, so I know how to calculate CDF1, CDF2 for every point.
I would like to find $x$ = the leftmost point where $\frac{CDF1(x)}{CDF2(x)}=P$ where P is a constant desired SNR ratio, say P=3.
I have a file with some several thousands of such normal distribution pairs, so best would be if there could be some Excel formula I could use for each such pair, getting its relevant $x$.
Is there an anlytical solution I'm missing? If there's none, can you point me to some nice way to do it with R/python without resulting to some monstrous "goal seek" Excel simulation?
Thanks! (Math and signal processing novice, be patient)
Consider $$\frac{CDF_1(x)}{CDF_2(x)} = P.$$
We have $$CDF(x) = \frac{1}{2} \left[1 + erf\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right]$$ and $$\frac{d}{dx}CDF(x) = \frac{1}{\sigma 2\sqrt{2}} e^{\frac{-(x-\mu)^2}{2\sigma^2}}.$$
Using the above two equations, write $$CDF_1(x) = P \times CDF_2(x)$$.
Differentiate on both sides with respect to $x$ and solve the ratio of exponentials using the log function. You should be getting a closed form expression for $x$.