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In his 1789 article, Bayes described a scenerio in which 6 balls are distributed independently and uniformly on a billiards table. A white ball defines a line, and you are told the count X of how many of n = 5 red balls end up to the left of that line. If we define the table to have length 1, we may assume the positions (left-to-right) of the balls are all distributed iid Uniform(0,1), with marking the position of the white ball and X counting the number of balls whose uniform variable ends up being less than 0. a) Given 0, what is the distribution of X, the count of uniform values less than 0? b) Find the marginal pmf for X, unconditional on 0. c) You are told X = 0. Find the conditional (posterior) distribution for given X = 0. Sketch the pdf for this distribtion.

In his 1789 article, Bayes described a scenerio in which 6 balls are distributed independently and uniformly on a billiards table. A white ball defines a line, and you are told the count X of how many of n = 5 red balls end up to the left of that line. If we define the table to have length 1, we may assume the positions (left-to-right) of the balls are all distributed iid Uniform(0,1), with marking the position of the white ball and X counting the number of balls whose uniform variable ends up being less than 0. a) Given 0, what is the distribution of X, the count of uniform values less than 0? b) Find the marginal pmf for X, unconditional on 0. c) You are told X = 0. Find the conditional (posterior) distribution for given X = 0. Sketch the pdf for this distribtion.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.2: The Least Square Line
Problem 2E
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In his 1789 article, Bayes described a scenerio in which 6 balls are distributed independently and uniformly on a billiards table. A white ball defines a line, and you are told the count X of how many of n = 5 red balls end up to the left of that line. If we define the table to have length 1, we may assume the positions (left-to-right) of the balls are all distributed iid Uniform(0,1), with 0 marking the position of the white ball and X counting the number of balls whose uniform variable ends up being less than 0. a) Given 0, what is the distribution of X, the count of uniform values less than 0? b) Find the marginal pmf for X, unconditional on 0. c) You are told X = 0. Find the conditional (posterior) distribution for 0 given X = 0. Sketch the pdf for this distribtion.
Transcribed Image Text:In his 1789 article, Bayes described a scenerio in which 6 balls are distributed independently and uniformly on a billiards table. A white ball defines a line, and you are told the count X of how many of n = 5 red balls end up to the left of that line. If we define the table to have length 1, we may assume the positions (left-to-right) of the balls are all distributed iid Uniform(0,1), with 0 marking the position of the white ball and X counting the number of balls whose uniform variable ends up being less than 0. a) Given 0, what is the distribution of X, the count of uniform values less than 0? b) Find the marginal pmf for X, unconditional on 0. c) You are told X = 0. Find the conditional (posterior) distribution for 0 given X = 0. Sketch the pdf for this distribtion.
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