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Does anyone know what the possible probability density function (pdf) are that $g(x)$ can take such that $f(x)/g(x)$ is non-decreasing, where $f(x)$ is a mixture of 2 normal densities? For example, take $$f(x)=\frac{1}{2}N(0,1)+\frac{1}{2}N(3,1).$$ $\int g(x)dx$ should be equal to $1$.

spaceisdarkgreen
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user340803
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  • What do you mean by mixture of two normal densities – Jonathan Davidson Jul 08 '17 at 05:59
  • Please use MathJax to format your equations. I have edited your post to be more readable. Click edit to see what I did to format it. It's easy... mostly you just enclose equations in dollar signs. – spaceisdarkgreen Jul 08 '17 at 06:04
  • I meant that f(x)=pф((x-μ_1)/σ_1 )+(1-p)ф((x-μ_2)/σ_2 ), where 0<p<1.

    Here’s some description on multimodal distribution. https://en.wikipedia.org/wiki/Multimodal_distribution

     – user340803 Jul 08 '17 at 06:13
  • Just to clarify, μ_1 and σ_1 are the mean and standard deviation of the first normal distribution; while μ_2 and σ_2 are the mean and standard deviation of second distribution. – user340803 Jul 08 '17 at 06:17
  • Is $g(x)=f(x)$ valid? $f(x)/g(x)=1$ is non-decreasing and $g(x)$ inegrates to $1$. – NCh Jul 09 '17 at 08:39

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