# Difference between revisions of "Eleisha's Segment 17: The Multivariate Normal Distribution"

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Hint: The answers are all simple rationals, but I had to use Mathematica to work them out. | Hint: The answers are all simple rationals, but I had to use Mathematica to work them out. | ||

+ | |||

+ | <b> To Think About: </b> | ||

+ | |||

+ | 1. Prove the assertions in slide 5. That is, implement the ideas in the blue text. | ||

+ | |||

+ | 2. How would you plot an error ellipsoid in 3 dimensions? That is, what would be the 3-dimensional version of the code in slide 8? (You can assume the plotting capabilities of your favorite programming language.) | ||

+ | |||

+ | <b> Back To: </b> [[Eleisha Jackson]] |

## Latest revision as of 10:04, 25 February 2014

** To Calculate: **

1. Calculate the Jacobian determinant of the transformation of variables defined by

**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 y_1 = x_1/x_2, \qquad y_2 = x_2^2 }**

2. Consider the 3-dimensional multivariate normal 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 (x_1,x_2,x_3) }**
with **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,-1,-1) }**
and

**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^{-1} = \left( \begin{array}{ccc} 5 & -1 & 2 \\ -1 & 8 & 1 \\ 2 & 1 & 4 \end{array}\right).}**
(Note the matrix inverse notation.)

What are 2-dimensional **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 }**
and **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^{-1} }**
for

(a) the distribution on the slice**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_3 =0 }**
?

(b) the marginalization 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 x_3 }**
?

Hint: The answers are all simple rationals, but I had to use Mathematica to work them out.

** To Think About: **

1. Prove the assertions in slide 5. That is, implement the ideas in the blue text.

2. How would you plot an error ellipsoid in 3 dimensions? That is, what would be the 3-dimensional version of the code in slide 8? (You can assume the plotting capabilities of your favorite programming language.)

** Back To: ** Eleisha Jackson