Correlation and mixture Suppose we have a mixture distribution (same parametric family) and we know that a third variable $X$ is correlated to each of those variables in the mixture with coefficients $\rho_1$, $\rho_2$, respectively. What is the correlation coefficient of $X$ with the mixture?

Since
$$Y=\begin{cases}X_1 &\text{with probability }\rho_1\\ X_2&\text{with probability }\rho_2=1-\rho_1\end{cases}$$
one can rewrite $Y$ as
$$Y=Z\times X_1+(1-Z)\times X_2$$
where $Z\sim\mathcal B(\rho_1)$. We will assume that $Z$ and $X$ are independent. Then
$$\mathbb E[Y]=\mathbb E[Z\times X_1+(1-Z)\times X_2]=\rho_1\mathbb E[X_1]+\rho_2\mathbb E[X_2]$$
and
\begin{align*}\mathbb E[XY]&=\rho_1\mathbb E[XX_1]+\rho_2\mathbb E[XX_2]\\
&=\rho_1\{\tau_1+\mathbb E[X]\mathbb E[X_1]\}+\rho_2\{\tau_2+\mathbb E[X]\mathbb E[X_2]\}\\
&= \rho_1\tau_1+\rho_2\tau_2 +\mathbb E[X]\{\rho_1\mathbb E[X_1]+\rho_2\mathbb E[X_2]\}
\end{align*}
Hence,
$$\text{cov}(X,Y)=\rho_1\tau_1+\rho_2\tau_2$$
