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

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+ | <b> To Calculate: </b> | ||

+ | |||

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

− | < | + | <math> y_1 = x_1/x_2, \qquad y_2 = x_2^2 </math> |

+ | |||

+ | 2. Consider the 3-dimensional multivariate normal over <math> (x_1,x_2,x_3) </math> with <math> \mu = (-1,-1,-1) </math> and | ||

<math> | <math> | ||

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-1 & 8 & 1 \\ | -1 & 8 & 1 \\ | ||

2 & 1 & 4 | 2 & 1 & 4 | ||

− | \end{array} | + | \end{array}\right).</math> (Note the matrix inverse notation.) |

− | \right).</math> (Note the matrix inverse notation.) | ||

What are 2-dimensional <math>\mu </math> and <math>\Sigma^{-1} </math> for | What are 2-dimensional <math>\mu </math> and <math>\Sigma^{-1} </math> for | ||

+ | |||

(a) the distribution on the slice<math> x_3 =0 </math>? | (a) the distribution on the slice<math> x_3 =0 </math>? | ||

+ | |||

(b) the marginalization over <math>x_3 </math> ? | (b) the marginalization over <math>x_3 </math> ? | ||

+ | |||

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. |

## Revision as of 10:02, 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.