Segment 8. Some Standard Distributions

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{{#widget:Iframe |url=http://www.youtube.com/v/EDYDC7iNGTg&hd=1 |width=800 |height=625 |border=0 }}

The direct YouTube link is http://youtu.be/EDYDC7iNGTg Links to the slides: PDF file or PowerPoint file

Problems

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 ExponentialFailed 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?)

Class Activity

problem statement: ClassActivity20130204.pdf

data file for class activity: events20130204.txt

Jeff's solution (Python)

Bill's solution (MATLAB)