exercise:1593654f61: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> A coin has an unknown bias <math>p</math> that is assumed to be uniformly distributed between 0 and 1. The coin is tossed <math>n</math> times and heads turns up <math>j</math> times and tails turns up <math>k</math> times. We have seen that the...") |
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A coin has an unknown bias <math>p</math> that is assumed to be uniformly distributed between 0 and 1. The coin is tossed <math>n</math> times and heads turns up <math>j</math> times and tails turns up <math>k</math> times. We have seen that the probability that | |||
distributed between 0 and 1. The coin is tossed <math>n</math> times and heads turns up | |||
<math>j</math> times and tails turns up <math>k</math> times. We have seen that the probability that | |||
heads turns up next time is | heads turns up next time is | ||
| Line 13: | Line 5: | ||
\frac {j + 1}{n + 2}\ . | \frac {j + 1}{n + 2}\ . | ||
</math> | </math> | ||
Show that this is the same as the probability that the next ball is black for | |||
the Polya urn model of | Show that this is the same as the probability that the next ball is black for the Polya urn model of [[exercise:7f19badaad|Exercise]]. Use this result to explain why, in the Polya urn model, the proportion of black balls does not tend to 0 or 1 as one might expect but rather to a uniform | ||
Use this result to explain why, in the Polya urn model, the proportion of black | |||
balls does not tend to 0 or 1 as one might expect but rather to a uniform | |||
distribution on the interval <math>[0,1]</math>. | distribution on the interval <math>[0,1]</math>. | ||
Latest revision as of 23:43, 13 June 2024
A coin has an unknown bias [math]p[/math] that is assumed to be uniformly distributed between 0 and 1. The coin is tossed [math]n[/math] times and heads turns up [math]j[/math] times and tails turns up [math]k[/math] times. We have seen that the probability that heads turns up next time is
[[math]]
\frac {j + 1}{n + 2}\ .
[[/math]]
Show that this is the same as the probability that the next ball is black for the Polya urn model of Exercise. Use this result to explain why, in the Polya urn model, the proportion of black balls does not tend to 0 or 1 as one might expect but rather to a uniform distribution on the interval [math][0,1][/math].