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Consider the following mixture of two densities $$ f(x)=\lambda g(x-\mu_1)+(1-\lambda)g(x-\mu_2) $$ with $\lambda\in [0,1]$, $g(\cdot)$ symmetric around zero, $\mu_1<\mu_2$.

Claim: the mixture is symmetric if and only if $\lambda\in \{0,1,\frac{1}{2}\}$.

Could you help me to show this? I understand that $\lambda\in \{0,1\}$ implies that the mixture is symmetric because $g(\cdot)$ is symmetric. What I'm struggling to understand is why the weights should be equal to get a symmetric distribution.

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    $\begingroup$ find whether or not there is a $\bar\mu$ such that$$f(x-\bar\mu)=f(\bar\mu-x)\quad\forall x$$ $\endgroup$ – Xi'an Feb 6 '19 at 14:32
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    $\begingroup$ Directly apply the definition at stats.stackexchange.com/questions/28992. $\endgroup$ – whuber Feb 6 '19 at 21:40

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