# How to Solve For the Inverse Cumulative Distribution Function of a Double-Exponential Probability Density Function

I'm stuck on figuring out how to sample data from a fake/known double-exponential PDF for a lab project involving C. elegans egg-laying rate data. I need help with figuring out if there's an exact solution for the inverse CDF for the double-exponential PDF I'm working with. Or, I could use advice on how to proceed with finding an analytical solution or alternative method for sampling from the PDF if there is no exact solution for the inverse CDF.

In the paper "A Three-State Biological Point Process Model and Its Parameter Estimation" by Zhou et. al., the experimenters found that a 3-parameter double-exponential PDF best models the interval times between egg lay events in C. elegans, given that C. elegans tend to lay a bunch of eggs in short bouts with longer periods of waiting time between bouts.

The PDF they came up with is: $$f(x)=k_1\lambda_1e^{(-\lambda_1x)} + k_2\lambda_2e^{(-p\lambda_2x)}$$, where $$k_1 = \frac{p(\lambda_1 - \lambda_2)}{\lambda_1 - p\lambda_2}$$ and $$k_2 = \frac{\lambda_1(1 - p)}{\lambda_1 - p\lambda_2}$$.

On page 7 of the paper, in footnote 5, the authors state that the CDF is: $$F(x)=1-k_1e^{(-\lambda_1x)}-k_2e^{(-p\lambda_2x)}$$

Is there an exact solution for the inverse of the CDF, $$F^{-1}(x)$$?

After attempting to derive an exact solution for $$F^{-1}(x)$$ myself, I can't find a way to isolate $$x$$, so I was wondering if there is some trick I don't know/can't think of. Or if there's a way to simplify the problem in polar coordinates or on the complex plane. For instance, could one switch the problem to polar coordinates, similar to the Box-Muller method for sampling from a Gaussian distribution?