# 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (b_1,b_2)}**
with 2-vector mean **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mu_1,\mu_2)}**
and 2x2 matrix covariance **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma}**
. What is the distribution of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_1}**
given that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_2}**
has the particular value **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_c}**
? In particular, what is the mean and standard deviation of the conditional distribution of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_2}**
instead of conditioning on it.

#### To Think About

1. Why should it be called the Fisher *Information* Matrix? What does it have to do with "information"?

2. Go read (e.g., in Wikipedia or elsewhere) about the "Cramer-Rao bound" and be prepared to explain what it is, and what it has to do with the Fisher Information Matrix.

### Class Activity

Today we'll do Find the Volcano.