I have found that if X has a weighted pdf then $E(X)$ is just the weighted $E(X_i)$'s and the same holds for the second moments. Can I cleanly express Var(X)?
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3Yes, look for the _total variance formula_. – Dilip Sarwate May 20 '15 at 01:27
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A standard term for your "weighted PDF" is *mixture distribution.* That keyword helps find duplicates on this site. Although the duplicate's title refers to "Gaussians," the problem has nothing to do with the shapes of the distributions, so the answers to the duplicate also answer this question. – whuber May 20 '15 at 14:34