Polya’s urn model [duplicate]

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  • a problem on Polya's urn scheme

    2 answers



Polya’s urn model supposes that an urn initially contains $r$ red and $b$ blue balls. At each stage a ball is randomly selected from the urn and is then returned along with m other balls of the same color. Let $X_k$ be the number of red balls drawn in the first $k$ selections.



(a) Find $mathbbE[X_1]$.



(b) Find $mathbbE[X_2]$.



(c) Find $mathbbE[X_3]$.



(d) Conjecture the value of $mathbbE[X_k]$, and then verify your conjecture by a conditioning argument.



(e) Give an intuitive proof for your conjecture.




probability expectation




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marked as duplicate by Community♦ Jul 31 at 12:33


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  • What are your thoughts?
    – Ninja hatori
    Jul 30 at 6:52














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  • a problem on Polya's urn scheme

    2 answers



Polya’s urn model supposes that an urn initially contains $r$ red and $b$ blue balls. At each stage a ball is randomly selected from the urn and is then returned along with m other balls of the same color. Let $X_k$ be the number of red balls drawn in the first $k$ selections.



(a) Find $mathbbE[X_1]$.



(b) Find $mathbbE[X_2]$.



(c) Find $mathbbE[X_3]$.



(d) Conjecture the value of $mathbbE[X_k]$, and then verify your conjecture by a conditioning argument.



(e) Give an intuitive proof for your conjecture.




probability expectation




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marked as duplicate by Community♦ Jul 31 at 12:33


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • What are your thoughts?
    – Ninja hatori
    Jul 30 at 6:52












up vote
0
down vote

favorite









up vote
0
down vote

favorite












This question already has an answer here:



  • a problem on Polya's urn scheme

    2 answers



Polya’s urn model supposes that an urn initially contains $r$ red and $b$ blue balls. At each stage a ball is randomly selected from the urn and is then returned along with m other balls of the same color. Let $X_k$ be the number of red balls drawn in the first $k$ selections.



(a) Find $mathbbE[X_1]$.



(b) Find $mathbbE[X_2]$.



(c) Find $mathbbE[X_3]$.



(d) Conjecture the value of $mathbbE[X_k]$, and then verify your conjecture by a conditioning argument.



(e) Give an intuitive proof for your conjecture.




probability expectation




share|cite|improve this question














This question already has an answer here:



  • a problem on Polya's urn scheme

    2 answers



Polya’s urn model supposes that an urn initially contains $r$ red and $b$ blue balls. At each stage a ball is randomly selected from the urn and is then returned along with m other balls of the same color. Let $X_k$ be the number of red balls drawn in the first $k$ selections.



(a) Find $mathbbE[X_1]$.



(b) Find $mathbbE[X_2]$.



(c) Find $mathbbE[X_3]$.



(d) Conjecture the value of $mathbbE[X_k]$, and then verify your conjecture by a conditioning argument.



(e) Give an intuitive proof for your conjecture.





This question already has an answer here:



  • a problem on Polya's urn scheme

    2 answers





probability expectation





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 30 at 7:16









pointguard0

695517




695517









asked Jul 30 at 6:32









Leona Wu

1




1




marked as duplicate by Community♦ Jul 31 at 12:33


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.






marked as duplicate by Community♦ Jul 31 at 12:33


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • What are your thoughts?
    – Ninja hatori
    Jul 30 at 6:52
















  • What are your thoughts?
    – Ninja hatori
    Jul 30 at 6:52















What are your thoughts?
– Ninja hatori
Jul 30 at 6:52




What are your thoughts?
– Ninja hatori
Jul 30 at 6:52










1 Answer
1






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Clearly, for the distribution of $X_1$ is Bernoulli with parameter $p$ being equal to $fracrb + r$ and hence
$$
mathbbE X_1 = fracrb + r.
$$
In fact, the answer remains the same for $mathbbE X_n, n > 1$. See this link for detailed intuitive explanation.



P.S. I'm also marking this question as a duplicate due to the link provided above.






share|cite|improve this answer





















  • Thx! I haven't found that problem before due to my poor research skills lol
    – Leona Wu
    Jul 31 at 12:33

















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote













Clearly, for the distribution of $X_1$ is Bernoulli with parameter $p$ being equal to $fracrb + r$ and hence
$$
mathbbE X_1 = fracrb + r.
$$
In fact, the answer remains the same for $mathbbE X_n, n > 1$. See this link for detailed intuitive explanation.



P.S. I'm also marking this question as a duplicate due to the link provided above.






share|cite|improve this answer





















  • Thx! I haven't found that problem before due to my poor research skills lol
    – Leona Wu
    Jul 31 at 12:33














up vote
0
down vote













Clearly, for the distribution of $X_1$ is Bernoulli with parameter $p$ being equal to $fracrb + r$ and hence
$$
mathbbE X_1 = fracrb + r.
$$
In fact, the answer remains the same for $mathbbE X_n, n > 1$. See this link for detailed intuitive explanation.



P.S. I'm also marking this question as a duplicate due to the link provided above.






share|cite|improve this answer





















  • Thx! I haven't found that problem before due to my poor research skills lol
    – Leona Wu
    Jul 31 at 12:33












up vote
0
down vote










up vote
0
down vote









Clearly, for the distribution of $X_1$ is Bernoulli with parameter $p$ being equal to $fracrb + r$ and hence
$$
mathbbE X_1 = fracrb + r.
$$
In fact, the answer remains the same for $mathbbE X_n, n > 1$. See this link for detailed intuitive explanation.



P.S. I'm also marking this question as a duplicate due to the link provided above.






share|cite|improve this answer













Clearly, for the distribution of $X_1$ is Bernoulli with parameter $p$ being equal to $fracrb + r$ and hence
$$
mathbbE X_1 = fracrb + r.
$$
In fact, the answer remains the same for $mathbbE X_n, n > 1$. See this link for detailed intuitive explanation.



P.S. I'm also marking this question as a duplicate due to the link provided above.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 30 at 7:00









pointguard0

695517




695517











  • Thx! I haven't found that problem before due to my poor research skills lol
    – Leona Wu
    Jul 31 at 12:33
















  • Thx! I haven't found that problem before due to my poor research skills lol
    – Leona Wu
    Jul 31 at 12:33















Thx! I haven't found that problem before due to my poor research skills lol
– Leona Wu
Jul 31 at 12:33




Thx! I haven't found that problem before due to my poor research skills lol
– Leona Wu
Jul 31 at 12:33


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