Variance of Univariate Gaussian Mixture

Question

Let $\mathbb{P}_1,\dots,\mathbb{P}_n$ be univariate Gaussian measures with respective means $m_1,\dots,m_n \in \mathbb{R}$ and respective variances $\sigma_1,\dots,\sigma_n$. Let $r_1,\dots,r_n$ be numbers in $(0,1)$ which sum to $1$. Is the variance of a random variable distributed according to $\sum_{i=1}^n r_i\mathbb{P}_n$ equal to $\sum_{i=1}^n r_i^2 \sigma_i^2$?

Answer 1

Define indipendent random variable $X_i$ with distribution $\mathbb{P}_i$. Thus $\mathbb{E}[X_i]=m_i$ and $Var(X_i)=\sigma_i^2$.

Finally note that $Var(\sum_{i=1}^n r_iX_i)=\sum_{i=1}^n Var(r_iX_i)$ because they are uncorrelated (they are independent) and then you get $\sum_{i=1}^n Var(r_iX_i)=\sum_{i=1}^n r_i^2 \sigma_i^2$ also if you do not assume $r_1,...,r_n$ have sum $1$.