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

(3 intermediate revisions by the same user not shown) | |||

Line 2: | Line 2: | ||

<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. | + | 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 <math> {\rm Gamma}(\alpha,\beta) </math> has a single mode at <math> (\alpha-1)/\beta </math> when <math>\alpha \ge 1 </math>. | 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>. | ||

Line 11: | Line 11: | ||

<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)? | + | 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> |

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?) | 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?) | ||

+ | |||

+ | |||

+ | <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