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I have a bimodal distribution which is an equally weighted mixture of two normal distributions with known means and standard deviations. I can easily compute a two quantile values using the percent point function for each normal distribution, but how can I obtain the quantile value for the combined bimodal distribution?

Pierre Delecto
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  • Please explain the sense of "composed." Perhaps you are considering a mixture of Normal distributions? In that case, you need to know the mixture weights and, ultimately, you will need to do a numerical search for the solution. Please search our site for normal mixture quantile, which gives plenty of solutions. But if this is not a mixture, what is it?? – whuber Mar 04 '22 at 16:23
  • Yes this would be a mixture of Normal distributions. The distributions are weighted equally. I have edited the question. – Pierre Delecto Mar 04 '22 at 16:25
  • I found a good answer to your question and link to it in the close message above. – whuber Mar 04 '22 at 16:27
  • @Pierre see the first equation here; https://en.wikipedia.org/wiki/Mixture_distribution#Finite_and_countable_mixtures ($F(x) = ...$). This gives the cdf. Quantiles could be obtained reasonably easily from F by numerical root-finding (as with uniroot in R say), obtaining $x=F^{-1}(p)$ by solving the equation $F(x)-p=0$ for $x$ using a root finder. It's not too hard to even work out some bounds to supply if required. Since $F$ is easy to differentiate, root finders that rely on derivatives can also be used. – Glen_b Mar 05 '22 at 04:14

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