# A 2 component mixture is symmetric if and only if $\lambda\in \{0,1,\frac{1}{2}\}$

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.

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