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I want to further understand a topic discussed in this post: Intuition of convex combination of probability measures. In a class material I have the following statement:

If $\nu_1$ and $\nu_2$ are probability distributions and $\alpha\in[0,1]$, then a random variable $X$ distribution $\alpha\nu_1 + (1-\alpha)\nu_2$ iff there exists a random variable $Y\sim Ber(\alpha)$ such that $X$ has conditional distribution $\nu_1$ given $Y=1$ and conditional distribution $\nu_2$ given $Y=0$.

The backward implication is fairly straight forward, as highlighted in the mentioned post: $$P(X\in A) = P(X\in A\mid Y=1)P(Y=1) + P(X\in A\mid Y=0)P(Y=0) = (\alpha\nu_1 + (1-\alpha)\nu_2)(A).$$

I am stuck with the forward implication. It is clear to me that the choice of the variable $Y$ is essential (it can't just be any Bernoulli with parameter $\alpha$), but I can't figure out how to choose it.

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Let's illustrate this with the two distributions having densities $f_1(x)$ and $f_2(x)$ with the density for the mixture distribution being $\alpha f_1(x) + (1-\alpha)f_2(x)$.

Then you can say $$\mathbb P(Y=1 \mid X =x) = \frac{\alpha f_1(x) }{\alpha f_1(x) + (1-\alpha)f_2(x)}$$ which will give you the marginal distribution for $Y$ of $\mathbb P(Y=1)=\alpha$ as desired and you can deduce the conditional densities $f_{X\mid Y=1}(x)=f_1(x)$ and $f_{X\mid Y=0}(x)=f_2(x)$.

For more general distributions, you need to translate the conditional probability for $Y$ into the limit of the ratio of probabilities in a small neighbourhood around $x$ and get the more general result, but it is essentially the same.

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