Segment 21. Marginalize or Condition Uninteresting Fitted Parameters

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Problems

To Calculate

1. Consider a 2-dimensional multivariate normal distribution of the random variable $(b_1,b_2)$ with 2-vector mean $(\mu_1,\mu_2)$ and 2x2 matrix covariance $\Sigma$. What is the distribution of $b_1$ given that $b_2$ has the particular value $b_c$? In particular, what is the mean and standard deviation of the conditional distribution of $b_1$? (Hint, either see Wikipedia "Multivariate normal distribution" for the general case, or else just work out this special case.)

2. Same, but marginalize over $b_2$ instead of conditioning on it.