Marginal and Posterior Distributions
An election is being held. There are two candidates, A and B, and there are n
voters. The probability of voting for Candidate A varies by city. There are m cities, labeled
1, 2, . . . , m. The jth city has nj voters, so n1 + n2 + · · · + nm = n. Let Xj be the number
of people in the jth city who vote for Candidate A, with Xj
|pj ∼ Bin(nj
, pj ). To reflect our uncertainty about the probability of voting in each city, we treat p1, . . . , pm as r.v.s,
with prior distribution asserting that they are i.i.d. Unif(0, 1). Assume that X1, . . . , Xm are
independent, both unconditionally and conditional on p1, . . . , pm. Let X be the total number
of votes for Candidate A.
(a) Find the marginal distribution of X1 and the posterior distribution of p1|X1 = k1.
(b) Find E(X) and Var(X) in terms of n and s, where s = n1^2 + n2^2 + .... + nm^2.
How do I begin finding the marginal distribution of X1?
probability probability-distributions random-variables variance expected-value
asked Nov 29 at 20:56
Bob Dylan
111
add a comment |
An election is being held. There are two candidates, A and B, and there are n
voters. The probability of voting for Candidate A varies by city. There are m cities, labeled
1, 2, . . . , m. The jth city has nj voters, so n1 + n2 + · · · + nm = n. Let Xj be the number
of people in the jth city who vote for Candidate A, with Xj
|pj ∼ Bin(nj
, pj ). To reflect our uncertainty about the probability of voting in each city, we treat p1, . . . , pm as r.v.s,
with prior distribution asserting that they are i.i.d. Unif(0, 1). Assume that X1, . . . , Xm are
independent, both unconditionally and conditional on p1, . . . , pm. Let X be the total number
of votes for Candidate A.
(a) Find the marginal distribution of X1 and the posterior distribution of p1|X1 = k1.
(b) Find E(X) and Var(X) in terms of n and s, where s = n1^2 + n2^2 + .... + nm^2.
How do I begin finding the marginal distribution of X1?
probability probability-distributions random-variables variance expected-value
asked Nov 29 at 20:56
Bob Dylan
111
add a comment |
An election is being held. There are two candidates, A and B, and there are n
voters. The probability of voting for Candidate A varies by city. There are m cities, labeled
1, 2, . . . , m. The jth city has nj voters, so n1 + n2 + · · · + nm = n. Let Xj be the number
of people in the jth city who vote for Candidate A, with Xj
|pj ∼ Bin(nj
, pj ). To reflect our uncertainty about the probability of voting in each city, we treat p1, . . . , pm as r.v.s,
with prior distribution asserting that they are i.i.d. Unif(0, 1). Assume that X1, . . . , Xm are
independent, both unconditionally and conditional on p1, . . . , pm. Let X be the total number
of votes for Candidate A.
(a) Find the marginal distribution of X1 and the posterior distribution of p1|X1 = k1.
(b) Find E(X) and Var(X) in terms of n and s, where s = n1^2 + n2^2 + .... + nm^2.
How do I begin finding the marginal distribution of X1?
probability probability-distributions random-variables variance expected-value
asked Nov 29 at 20:56
Bob Dylan
111
An election is being held. There are two candidates, A and B, and there are n
voters. The probability of voting for Candidate A varies by city. There are m cities, labeled
1, 2, . . . , m. The jth city has nj voters, so n1 + n2 + · · · + nm = n. Let Xj be the number
of people in the jth city who vote for Candidate A, with Xj
|pj ∼ Bin(nj
, pj ). To reflect our uncertainty about the probability of voting in each city, we treat p1, . . . , pm as r.v.s,
with prior distribution asserting that they are i.i.d. Unif(0, 1). Assume that X1, . . . , Xm are
independent, both unconditionally and conditional on p1, . . . , pm. Let X be the total number
of votes for Candidate A.
(a) Find the marginal distribution of X1 and the posterior distribution of p1|X1 = k1.
(b) Find E(X) and Var(X) in terms of n and s, where s = n1^2 + n2^2 + .... + nm^2.
How do I begin finding the marginal distribution of X1?
probability probability-distributions random-variables variance expected-value
probability probability-distributions random-variables variance expected-value
asked Nov 29 at 20:56
Bob Dylan
111
asked Nov 29 at 20:56
Bob Dylan
111
asked Nov 29 at 20:56
Bob Dylan
111
asked Nov 29 at 20:56
Bob Dylan
111
asked Nov 29 at 20:56
Bob Dylan
111
111
add a comment |
add a comment |
1 Answer
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Hint for the marginal distribution of $X_1$:
$$mathbb P(X_1=k_1mid p_1)={n_1 choose k_1}p_1^{k_1}(1-p_1)^{n_1-k_1} text{ for } k_1 in {0,1,2,ldots,n_1}$$
$$f(p_1)=1text{ for } 0 le p_1 le 1$$
$$mathbb P(X_1=k_1)= int_{p_1} mathbb P(X_1=k_1mid p_1) f(p_1), dp_1$$
The answer should not be a big surprise given that you essentially know nothing about $p_1$
answered Nov 29 at 21:51
Henry
98k475160
add a comment |
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1 Answer
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active
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Hint for the marginal distribution of $X_1$:
$$mathbb P(X_1=k_1mid p_1)={n_1 choose k_1}p_1^{k_1}(1-p_1)^{n_1-k_1} text{ for } k_1 in {0,1,2,ldots,n_1}$$
$$f(p_1)=1text{ for } 0 le p_1 le 1$$
$$mathbb P(X_1=k_1)= int_{p_1} mathbb P(X_1=k_1mid p_1) f(p_1), dp_1$$
The answer should not be a big surprise given that you essentially know nothing about $p_1$
answered Nov 29 at 21:51
Henry
98k475160
add a comment |
Hint for the marginal distribution of $X_1$:
$$mathbb P(X_1=k_1mid p_1)={n_1 choose k_1}p_1^{k_1}(1-p_1)^{n_1-k_1} text{ for } k_1 in {0,1,2,ldots,n_1}$$
$$f(p_1)=1text{ for } 0 le p_1 le 1$$
$$mathbb P(X_1=k_1)= int_{p_1} mathbb P(X_1=k_1mid p_1) f(p_1), dp_1$$
The answer should not be a big surprise given that you essentially know nothing about $p_1$
answered Nov 29 at 21:51
Henry
98k475160
add a comment |
Hint for the marginal distribution of $X_1$:
$$mathbb P(X_1=k_1mid p_1)={n_1 choose k_1}p_1^{k_1}(1-p_1)^{n_1-k_1} text{ for } k_1 in {0,1,2,ldots,n_1}$$
$$f(p_1)=1text{ for } 0 le p_1 le 1$$
$$mathbb P(X_1=k_1)= int_{p_1} mathbb P(X_1=k_1mid p_1) f(p_1), dp_1$$
The answer should not be a big surprise given that you essentially know nothing about $p_1$
answered Nov 29 at 21:51
Henry
98k475160
Hint for the marginal distribution of $X_1$:
$$mathbb P(X_1=k_1mid p_1)={n_1 choose k_1}p_1^{k_1}(1-p_1)^{n_1-k_1} text{ for } k_1 in {0,1,2,ldots,n_1}$$
$$f(p_1)=1text{ for } 0 le p_1 le 1$$
$$mathbb P(X_1=k_1)= int_{p_1} mathbb P(X_1=k_1mid p_1) f(p_1), dp_1$$
The answer should not be a big surprise given that you essentially know nothing about $p_1$
answered Nov 29 at 21:51
Henry
98k475160
answered Nov 29 at 21:51
Henry
98k475160
answered Nov 29 at 21:51
Henry
98k475160
answered Nov 29 at 21:51
Henry
98k475160
98k475160
add a comment |
add a comment |
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