Mean and Variance of a Mixture of Normals
A random variable $X$ follows a two-component normal mixture: with probability $w$ it is drawn from $N(\mu_1, \sigma_1^2)$, and with probability
-w$ it is drawn from $N(\mu_2, \sigma_2^2)$.
1. Compute $E[X]$.
2. Compute $\text{Var}(X)$. Your answer should make clear why a mixture can have higher variance than either component alone.
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