Mixing
Describe the statistical model for the observed data ($T$) [closed]
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A random sample of $6$ observations $(X_1, X_2, cdots, X_6)$ is generated from a Geometric($theta$), where $theta in (0, 1)$ unknown, but only $T = sum_{i=1}^{6} X_i$ is observed by the statistician.
(a) Describe the statistical model for the observed data ($T$)
(b) (i) Is it possible to parameterize the model by $Psi = frac{1-theta}{theta}$ ? Prove your answer
(ii) Is it possible to parameterize the model by $Psi = theta(1-theta)$ ? Prove your answer
probability statistical-inference
asked Jan 23 at 4:21
Bas basBas bas
49012
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closed as unclear what you're asking by Brian Borchers, Cesareo, Lee David Chung Lin, metamorphy, Gibbs Jan 23 at 21:04
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
A random sample of $6$ observations $(X_1, X_2, cdots, X_6)$ is generated from a Geometric($theta$), where $theta in (0, 1)$ unknown, but only $T = sum_{i=1}^{6} X_i$ is observed by the statistician.
(a) Describe the statistical model for the observed data ($T$)
(b) (i) Is it possible to parameterize the model by $Psi = frac{1-theta}{theta}$ ? Prove your answer
(ii) Is it possible to parameterize the model by $Psi = theta(1-theta)$ ? Prove your answer
probability statistical-inference
asked Jan 23 at 4:21
Bas basBas bas
49012
$endgroup$
closed as unclear what you're asking by Brian Borchers, Cesareo, Lee David Chung Lin, metamorphy, Gibbs Jan 23 at 21:04
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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Sum of i.i.d Geometric variables has a Negative binomial distribution. See math.stackexchange.com/questions/548525/….
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– StubbornAtom
Jan 23 at 8:13
$begingroup$
This post should NOT be reopened because it is asked again, arguably with improved content.
$endgroup$
– Lee David Chung Lin
Jan 25 at 4:12
add a comment |
$begingroup$
A random sample of $6$ observations $(X_1, X_2, cdots, X_6)$ is generated from a Geometric($theta$), where $theta in (0, 1)$ unknown, but only $T = sum_{i=1}^{6} X_i$ is observed by the statistician.
(a) Describe the statistical model for the observed data ($T$)
(b) (i) Is it possible to parameterize the model by $Psi = frac{1-theta}{theta}$ ? Prove your answer
(ii) Is it possible to parameterize the model by $Psi = theta(1-theta)$ ? Prove your answer
probability statistical-inference
asked Jan 23 at 4:21
Bas basBas bas
49012
$endgroup$
A random sample of $6$ observations $(X_1, X_2, cdots, X_6)$ is generated from a Geometric($theta$), where $theta in (0, 1)$ unknown, but only $T = sum_{i=1}^{6} X_i$ is observed by the statistician.
(a) Describe the statistical model for the observed data ($T$)
(b) (i) Is it possible to parameterize the model by $Psi = frac{1-theta}{theta}$ ? Prove your answer
(ii) Is it possible to parameterize the model by $Psi = theta(1-theta)$ ? Prove your answer
probability statistical-inference
probability statistical-inference
asked Jan 23 at 4:21
Bas basBas bas
49012
asked Jan 23 at 4:21
Bas basBas bas
49012
edited Jan 25 at 3:31
Bas bas
asked Jan 23 at 4:21
Bas basBas bas
49012
asked Jan 23 at 4:21
Bas basBas bas
49012
asked Jan 23 at 4:21
Bas basBas bas
49012
49012
closed as unclear what you're asking by Brian Borchers, Cesareo, Lee David Chung Lin, metamorphy, Gibbs Jan 23 at 21:04
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Brian Borchers, Cesareo, Lee David Chung Lin, metamorphy, Gibbs Jan 23 at 21:04
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Sum of i.i.d Geometric variables has a Negative binomial distribution. See math.stackexchange.com/questions/548525/….
$endgroup$
– StubbornAtom
Jan 23 at 8:13
$begingroup$
This post should NOT be reopened because it is asked again, arguably with improved content.
$endgroup$
– Lee David Chung Lin
Jan 25 at 4:12
add a comment |
$begingroup$
Sum of i.i.d Geometric variables has a Negative binomial distribution. See math.stackexchange.com/questions/548525/….
$endgroup$
– StubbornAtom
Jan 23 at 8:13
$begingroup$
This post should NOT be reopened because it is asked again, arguably with improved content.
$endgroup$
– Lee David Chung Lin
Jan 25 at 4:12
$begingroup$
Sum of i.i.d Geometric variables has a Negative binomial distribution. See math.stackexchange.com/questions/548525/….
$endgroup$
– StubbornAtom
Jan 23 at 8:13
$begingroup$
Sum of i.i.d Geometric variables has a Negative binomial distribution. See math.stackexchange.com/questions/548525/….
$endgroup$
– StubbornAtom
Jan 23 at 8:13
$begingroup$
This post should NOT be reopened because it is asked again, arguably with improved content.
$endgroup$
– Lee David Chung Lin
Jan 25 at 4:12
$begingroup$
This post should NOT be reopened because it is asked again, arguably with improved content.
$endgroup$
– Lee David Chung Lin
Jan 25 at 4:12
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Watch out! The fact that $X_i sim Geo(theta)$ by no means implies that the sum of i.i.d. geometric random variables (i.e. $T$) has a geometric distribution!
Recall that the terminology random sample implies that the observations have the distribution that you have mentioned and are also conditionally independent.
Recall also the definition of statistical model for random samples.
Considering the random sample $mathbf{X}_n={X_1,dots,X_n}$, let $mathcal{X}$ be the sample space for one of the random variables and $Theta$ the parameter space. Any statistical model for random samples has the following form
$$left{mathcal{X^n} quad; quad f_n(mathbf{x}_n;theta) = prod_{i=1}^n f_{X}(x_i;theta) quad ; quad thetainThetaright}$$
where $f_X$ is the probability law of one of the random variables in the random sample and $f_n$ is the joint probability of the random vector $mathbf{X}_n$. That is, a triplet consisting of a sample space (for the random vector $mathbf{X}_n$), a joint parametric probability distribution and a parameter space on which the family of distributions is parametrized.
So, steps to solve your problem:
1) Identify the correct sample space for your random variable $T$.
2) Determine the joint distribution of $mathbf{X}_n$ by considering the independence and identical distribution of the random variables $X_1, X_2, dots, X_6$.
3) Consider that $T$ is simply a statistic of the random sample, and so, from your statistical model for the random vector, you can easily find the statistical model for a function of the random vector. Just remember what you are looking for. A sample space (recall that $T: mathcal{X}^nrightarrow mathcal{T}$, with $mathcal{T}$ being the space in which the statistic $T$ has values), a probability law (this time for the sum of random geometric variables) and a parameter space.
answered Jan 23 at 7:39
Easymode44Easymode44
417212
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Watch out! The fact that $X_i sim Geo(theta)$ by no means implies that the sum of i.i.d. geometric random variables (i.e. $T$) has a geometric distribution!
Recall that the terminology random sample implies that the observations have the distribution that you have mentioned and are also conditionally independent.
Recall also the definition of statistical model for random samples.
Considering the random sample $mathbf{X}_n={X_1,dots,X_n}$, let $mathcal{X}$ be the sample space for one of the random variables and $Theta$ the parameter space. Any statistical model for random samples has the following form
$$left{mathcal{X^n} quad; quad f_n(mathbf{x}_n;theta) = prod_{i=1}^n f_{X}(x_i;theta) quad ; quad thetainThetaright}$$
where $f_X$ is the probability law of one of the random variables in the random sample and $f_n$ is the joint probability of the random vector $mathbf{X}_n$. That is, a triplet consisting of a sample space (for the random vector $mathbf{X}_n$), a joint parametric probability distribution and a parameter space on which the family of distributions is parametrized.
So, steps to solve your problem:
1) Identify the correct sample space for your random variable $T$.
2) Determine the joint distribution of $mathbf{X}_n$ by considering the independence and identical distribution of the random variables $X_1, X_2, dots, X_6$.
3) Consider that $T$ is simply a statistic of the random sample, and so, from your statistical model for the random vector, you can easily find the statistical model for a function of the random vector. Just remember what you are looking for. A sample space (recall that $T: mathcal{X}^nrightarrow mathcal{T}$, with $mathcal{T}$ being the space in which the statistic $T$ has values), a probability law (this time for the sum of random geometric variables) and a parameter space.
answered Jan 23 at 7:39
Easymode44Easymode44
417212
$endgroup$
add a comment |
$begingroup$
Watch out! The fact that $X_i sim Geo(theta)$ by no means implies that the sum of i.i.d. geometric random variables (i.e. $T$) has a geometric distribution!
Recall that the terminology random sample implies that the observations have the distribution that you have mentioned and are also conditionally independent.
Recall also the definition of statistical model for random samples.
Considering the random sample $mathbf{X}_n={X_1,dots,X_n}$, let $mathcal{X}$ be the sample space for one of the random variables and $Theta$ the parameter space. Any statistical model for random samples has the following form
$$left{mathcal{X^n} quad; quad f_n(mathbf{x}_n;theta) = prod_{i=1}^n f_{X}(x_i;theta) quad ; quad thetainThetaright}$$
where $f_X$ is the probability law of one of the random variables in the random sample and $f_n$ is the joint probability of the random vector $mathbf{X}_n$. That is, a triplet consisting of a sample space (for the random vector $mathbf{X}_n$), a joint parametric probability distribution and a parameter space on which the family of distributions is parametrized.
So, steps to solve your problem:
1) Identify the correct sample space for your random variable $T$.
2) Determine the joint distribution of $mathbf{X}_n$ by considering the independence and identical distribution of the random variables $X_1, X_2, dots, X_6$.
3) Consider that $T$ is simply a statistic of the random sample, and so, from your statistical model for the random vector, you can easily find the statistical model for a function of the random vector. Just remember what you are looking for. A sample space (recall that $T: mathcal{X}^nrightarrow mathcal{T}$, with $mathcal{T}$ being the space in which the statistic $T$ has values), a probability law (this time for the sum of random geometric variables) and a parameter space.
answered Jan 23 at 7:39
Easymode44Easymode44
417212
$endgroup$
add a comment |
$begingroup$
Watch out! The fact that $X_i sim Geo(theta)$ by no means implies that the sum of i.i.d. geometric random variables (i.e. $T$) has a geometric distribution!
Recall that the terminology random sample implies that the observations have the distribution that you have mentioned and are also conditionally independent.
Recall also the definition of statistical model for random samples.
Considering the random sample $mathbf{X}_n={X_1,dots,X_n}$, let $mathcal{X}$ be the sample space for one of the random variables and $Theta$ the parameter space. Any statistical model for random samples has the following form
$$left{mathcal{X^n} quad; quad f_n(mathbf{x}_n;theta) = prod_{i=1}^n f_{X}(x_i;theta) quad ; quad thetainThetaright}$$
where $f_X$ is the probability law of one of the random variables in the random sample and $f_n$ is the joint probability of the random vector $mathbf{X}_n$. That is, a triplet consisting of a sample space (for the random vector $mathbf{X}_n$), a joint parametric probability distribution and a parameter space on which the family of distributions is parametrized.
So, steps to solve your problem:
1) Identify the correct sample space for your random variable $T$.
2) Determine the joint distribution of $mathbf{X}_n$ by considering the independence and identical distribution of the random variables $X_1, X_2, dots, X_6$.
3) Consider that $T$ is simply a statistic of the random sample, and so, from your statistical model for the random vector, you can easily find the statistical model for a function of the random vector. Just remember what you are looking for. A sample space (recall that $T: mathcal{X}^nrightarrow mathcal{T}$, with $mathcal{T}$ being the space in which the statistic $T$ has values), a probability law (this time for the sum of random geometric variables) and a parameter space.
answered Jan 23 at 7:39
Easymode44Easymode44
417212
$endgroup$
Watch out! The fact that $X_i sim Geo(theta)$ by no means implies that the sum of i.i.d. geometric random variables (i.e. $T$) has a geometric distribution!
Recall that the terminology random sample implies that the observations have the distribution that you have mentioned and are also conditionally independent.
Recall also the definition of statistical model for random samples.
Considering the random sample $mathbf{X}_n={X_1,dots,X_n}$, let $mathcal{X}$ be the sample space for one of the random variables and $Theta$ the parameter space. Any statistical model for random samples has the following form
$$left{mathcal{X^n} quad; quad f_n(mathbf{x}_n;theta) = prod_{i=1}^n f_{X}(x_i;theta) quad ; quad thetainThetaright}$$
where $f_X$ is the probability law of one of the random variables in the random sample and $f_n$ is the joint probability of the random vector $mathbf{X}_n$. That is, a triplet consisting of a sample space (for the random vector $mathbf{X}_n$), a joint parametric probability distribution and a parameter space on which the family of distributions is parametrized.
So, steps to solve your problem:
1) Identify the correct sample space for your random variable $T$.
2) Determine the joint distribution of $mathbf{X}_n$ by considering the independence and identical distribution of the random variables $X_1, X_2, dots, X_6$.
3) Consider that $T$ is simply a statistic of the random sample, and so, from your statistical model for the random vector, you can easily find the statistical model for a function of the random vector. Just remember what you are looking for. A sample space (recall that $T: mathcal{X}^nrightarrow mathcal{T}$, with $mathcal{T}$ being the space in which the statistic $T$ has values), a probability law (this time for the sum of random geometric variables) and a parameter space.
answered Jan 23 at 7:39
Easymode44Easymode44
417212
edited Jan 23 at 7:51
answered Jan 23 at 7:39
Easymode44Easymode44
417212
answered Jan 23 at 7:39
Easymode44Easymode44
417212
answered Jan 23 at 7:39
Easymode44Easymode44
417212
417212
add a comment |
add a comment |
$begingroup$
Sum of i.i.d Geometric variables has a Negative binomial distribution. See math.stackexchange.com/questions/548525/….
$endgroup$
– StubbornAtom
Jan 23 at 8:13
$begingroup$
This post should NOT be reopened because it is asked again, arguably with improved content.
$endgroup$
– Lee David Chung Lin
Jan 25 at 4:12