Mixing
Polya’s urn model [duplicate]
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a problem on Polya's urn scheme
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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
edited Jul 30 at 7:16
pointguard0
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asked Jul 30 at 6:32
Leona Wu
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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.
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up vote
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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
edited Jul 30 at 7:16
pointguard0
695517
asked Jul 30 at 6:32
Leona Wu
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.
What are your thoughts?
– Ninja hatori
Jul 30 at 6:52
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up vote
0
down vote
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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
edited Jul 30 at 7:16
pointguard0
695517
asked Jul 30 at 6:32
Leona Wu
1
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
edited Jul 30 at 7:16
pointguard0
695517
asked Jul 30 at 6:32
Leona Wu
1
edited Jul 30 at 7:16
pointguard0
695517
edited Jul 30 at 7:16
pointguard0
695517
edited Jul 30 at 7:16
pointguard0
695517
695517
asked Jul 30 at 6:32
Leona Wu
1
asked Jul 30 at 6:32
Leona Wu
1
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
add a comment |Â
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
add a comment |Â
1 Answer
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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.
answered Jul 30 at 7:00
pointguard0
695517
Thx! I haven't found that problem before due to my poor research skills lol
– Leona Wu
Jul 31 at 12:33
add a comment |Â
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.
answered Jul 30 at 7:00
pointguard0
695517
Thx! I haven't found that problem before due to my poor research skills lol
– Leona Wu
Jul 31 at 12:33
add a comment |Â
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.
answered Jul 30 at 7:00
pointguard0
695517
Thx! I haven't found that problem before due to my poor research skills lol
– Leona Wu
Jul 31 at 12:33
add a comment |Â
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.
answered Jul 30 at 7:00
pointguard0
695517
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.
answered Jul 30 at 7:00
pointguard0
695517
answered Jul 30 at 7:00
pointguard0
695517
answered Jul 30 at 7:00
pointguard0
695517
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
add a comment |Â
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
add a comment |Â
What are your thoughts?
– Ninja hatori
Jul 30 at 6:52