Eleisha's Segment 8: Some Standard Distributions

From Computational Statistics Course Wiki
Revision as of 13:06, 5 February 2014 by Eleishaj (talk | contribs)
Jump to navigation Jump to search

To Calculate

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.

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, P(x). How would you design an algorithm to compute its inverse (see slide 9) x(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 } )?

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?)