# Segment 8. Some Standard Distributions

#### Watch this segment

(Don't worry, what you see statically below is not the beginning of the segment. Press the play button to start at the beginning.)

{{#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 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?)

### Class Activity

problem statement: ClassActivity20130204.pdf

data file for class activity: events20130204.txt