In the basic Pólya urn model, the urn contains x white and y black balls; one ball is drawn randomly from the urn and its color observed; it is then replaced in the urn, and an additional ball of the same color is added to the urn.

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Probability of every ball occurring in multiple independent random samples

An urn contains 5 distinct numbered balls. You choose 2 without replacement. You then reset the urn and choose another 2 without replacement. Do this one more time. Now you have three random samples ...
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Distribution in Polya's Urn / existence of mgf / Stolz–Cesàro alternative / dominated convergence theorem

I know this has been asked elsewhere, but I think the values or random variables are different or something. From Williams' Probability with Martingales: I proved that $M_n$ is a $\sigma(B_1, ...
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59 views

References for a Variant of Polya's Urn Scheme

A row of $n_1$ balls, all coloured with colour #1, is laid out before me. I do a set of $k$ replacements (where $k<n_1$) in steps: Step 1: Choose a (uniform) random number $n_2$ on the range ...
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Polya's Urn : proportion Probability

There are two urns with one ball each. Each of subsequent n-2 balls is placed into one of these urns, with probability proportional to the number of balls already in that urn. What is the expected ...
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Expectation of longst run of balls in a Polya-like urn?

We have an urn with $N$ black balls. A single ball is drawn. If the ball is black, it is replaced with a white ball. If the ball is white, it is not replaced. We repeat draws until the urn is ...
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35 views

Consecutive Draws from Urn

An urn contains 4 black balls and 6 white balls. What is the probability of getting 1 black ball and 1 white ball in two consecutive draws from the urn? I know this is supposed to be easy. So i tried ...
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Discrete time Pure-Birth process with population fraction

I would like to solve difference equations for a pure-birth process, where the rate of adding new nodes depends on the fraction of population. $$x_i(t+1)-x_i(t)=\alpha ...
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75 views

Urn problem: with and without replacement the probability of seeing last ball of particular color

suppose you have red and blue balls in a jar and you pick one at a time, if its red you replace it, additionally you do another draw and throw whichever ball comes up. if its blue you throw it ...
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32 views

Counting Operations in the context of an Urn Problem

I was tasked with the following question, regarding the counting of operations in the pseudo code provided that has nested for loops: Let U ={B1,B2,...,Bn} with n >= 3. Interpret the following ...
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1answer
49 views

Resampling Probability

I've had a student working on some resampling problems. Here's a question I've posed to her: M consecutive draws of size K are taken from an urn with N balls. Balls are replaced between draws. ...
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Formula for urn model with replacement and “color switching”

I tried to figure out following problem, but failed :/ My best guess is that it somehow is a mixture of a binomial and a hypergeometric distribution: I have an urn with N white balls and 0 black ...
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1answer
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Expected number of red balls removed from an urn before the first black ball

Question: An urn contains n+m balls of which n are red and m are black. They are withdrawn from the urn one at a time and without replacement. Let $X$ be the number of red balls removed before the ...
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Polya's urn model - limit distribution

Let an urn contain w white and b black balls. Draw a ball randomly from the urn and return it together with another ball of the same color. Let $b_n$ be the number of black balls and $w_n$ the number ...
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Multicolored Polya Eggenberger Urn distributions

From an urn containing 1 ball each of $n$ different colours, a ball is drawn, its colour noted and then it is returned to the urn along with $k$ balls of the same colour. what is the limit of the ...
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66 views

Number of draws to see every object at least once

I woke up this morning with the following question on my mind: What is (the distribution of) the number of draws you need to sample (with replacement) from an urn with $n$ distinct objects until you ...
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128 views

Equilibrium distributions for a finite urn scheme

Given an urn with $n$ (fixed) balls that can be red or black, and given two parameters $0 < p, \, q < 1$, keep doing the following: Flip a $p$-coin. If heads come up, remove a black ball or ...