# Segment 21 Sanmit Narvekar

## Segment 21

#### 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.)

According to WIkipedia, the conditional distribution of X_1 given X_2 is (note this is mean and variance, not standard deviation):

**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 X_1|X_2=x_2 \ \sim\ \mathcal{N}\left(\mu_1+\frac{\sigma_1}{\sigma_2}\rho( x_2 - \mu_2),\, (1-\rho^2)\sigma_1^2\right). }**

where

**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 \rho_{X,Y}={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} }**

Thus, we can calculate the conditional mean and standard deviation as follows:

**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 \rho = \frac{\Sigma_{12}}{\sqrt{\Sigma_{11}\Sigma_{22}}}}**

**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 = \mu_1 + \frac{\sqrt{\Sigma_{11}}}{\sqrt{\Sigma_{22}}} \frac{\Sigma_{12}}{\sqrt{\Sigma_{11} \Sigma_{22}}} (b_c - \mu_2) = \mu_1 + \frac{\Sigma_{12}}{\Sigma_{22}} (b_c - \mu_2)}**

**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 = \sqrt{ \left( 1 - \frac{\Sigma_{12}^2}{\Sigma_{11}\Sigma_{22}} \right) \Sigma_{11} } = \sqrt{ \frac{\Sigma_{22}\Sigma_{11} - \Sigma_{12}^2}{\Sigma_{22}}}}**

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 marginalize over b2, we simply remove those rows/columns from the covariance matrix, and also from the mean vector. Thus, the marginalized mean and standard deviation are:

**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 = \mu_1}**

**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 = \sqrt{\Sigma_{11}}}**

#### 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.