Mean, Variance, and Bimodality of a Gaussian Mixture

A random variable $X$ is drawn from a mixture of two normals: with probability $p$ we sample from $N(\mu_1, \sigma_1^2)$ and with probability $1-p$ from $N(\mu_2, \sigma_2^2)$.

(a) Derive $E[X]$ and $\text{Var}(X)$ in terms of $p, \mu_1, \mu_2, \sigma_1^2, \sigma_2^2$.

(b) For the symmetric case $p = 1/2$ and $\sigma_1 = \sigma_2 = \sigma$, show that the mixture PDF is bimodal if and only if $|\mu_1 - \mu_2| > 2\sigma$.

(c) Compute $E[X]$ and $\text{Var}(X)$ for $p = 1/2$, $\mu_1 = -2$, $\mu_2 = 2$, $\sigma_1 = \sigma_2 = 1$, and verify that this mixture is bimodal.