Difference between revisions of "Eleisha's Segment 8: Some Standard Distributions"

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<b> To Calculate </b>
 
<b> To Calculate </b>
  
1. In Segment 6 (slide 8) we used the improper prior 1/r. Show that this is just a limiting case of a (completely proper) Lognormal prior.
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1. In Segment 6 (slide 8) we used the improper prior <math> 1/r </math>. Show that this is just a limiting case of a (completely proper) Lognormal prior.
  
2. Prove that {\rm Gamma}(\alpha,\beta) has a single mode at (\alpha-1)/\beta when \alpha \ge 1.
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2. Prove that <math> {\rm Gamma}(\alpha,\beta) </math> has a single mode at <math> (\alpha-1)/\beta </math> when <math>\alpha \ge 1 </math>.
  
3. Show that the limiting case of the Student distribution as \nu\rightarrow\infty is the Normal distribution.
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3. Show that the limiting case of the Student distribution as <math> \nu\rightarrow\infty </math> is the Normal distribution.
  
  
 
<b> To Think About </b>
 
<b> To Think About </b>
1. Suppose you have an algorithm that can compute a CDF, P(x). How would you design an algorithm to compute its inverse (see slide 9) x(P)?
 
  
2. The lifetime t of a radioactive nucleus (say Uranium 238) is distributed as the Exponential distribution. Do you know why? (Hint: What is the distribution of an Exponential(\beta) random variable conditioned on its being greater than some given value?)
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1. Suppose you have an algorithm that can compute a CDF, <math> P(x) </math>. How would you design an algorithm to compute its inverse (see slide 9) <math> x(P)? </math>
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2. The lifetime t of a radioactive nucleus (say Uranium 238) is distributed as the Exponential distribution. Do you know why? (Hint: What is the distribution of an Exponential(<math> \beta </math>) random variable conditioned on its being greater than some given value?)
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<b> Back To: </b> [[Eleisha Jackson]]

Latest revision as of 10:45, 18 February 2014

To Calculate

1. In Segment 6 (slide 8) we used the improper prior 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 1/r } . Show that this is just a limiting case of a (completely proper) Lognormal prior.

2. Prove 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 {\rm Gamma}(\alpha,\beta) } has a single mode at 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 (\alpha-1)/\beta } when 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 \alpha \ge 1 } .

3. Show that the limiting case of the Student distribution as 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 \nu\rightarrow\infty } is the Normal distribution.


To Think About

1. Suppose you have an algorithm that can compute a CDF, 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 P(x) } . How would you design an algorithm to compute its inverse (see slide 9) 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(P)? }

2. The lifetime t of a radioactive nucleus (say Uranium 238) is distributed as the Exponential distribution. Do you know why? (Hint: What is the distribution of an Exponential(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 \beta } ) random variable conditioned on its being greater than some given value?)


Back To: Eleisha Jackson