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

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<b> To Calculate: </b>
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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>
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<math> y_1 = x_1/x_2, \qquad y_2 = x_2^2 </math>
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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}
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\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
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(a) the distribution on the slice<math> x_3 =0 </math>?
 
(a) the distribution on the slice<math> x_3 =0 </math>?
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(b) the marginalization over <math>x_3 </math> ?
 
(b) the marginalization over <math>x_3 </math> ?
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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.

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 sliceFailed 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.