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I have a mixture distribution of 2 Gaussians. Here, the left has a weight of 0.1 and the right has a weight of 0.9. In this example, they have identical $\sigma$, but that may not always be the case.

In my use case, the $\mu$, $\sigma$, and weights of the component distributions which comprise the mixture are always known.

Knowing the parameters of the component distributions, is there an analytical method for me to calculate the threshold probability density $D_{thresh}$, for which all density values $D \ge D_{thresh}$ constitute the region of 95% cumulative probability in this mixed distribution?

enter image description here

Red lines indicate the +/- 1.96 SD for each component distribution.

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