Say that I have X|θ ∼ N(θ, 1) and θ ∼ N(0, 1). I know the mean of X is 0, but how do I calculate its variance? My guess is that its the variance of both normal distributions added together, but I'm not sure why exactly.
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See https://stats.stackexchange.com/questions/16608/what-is-the-variance-of-the-weighted-mixture-of-two-gaussians – kjetil b halvorsen Dec 05 '20 at 19:27
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You can use law of total variance: $$\operatorname{var}(X)=\mathbb E[\operatorname{var}(X|\theta)]+\operatorname{var}(\mathbb E[X|\theta])=1+1=2$$
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