The problem is asking to show that the sum of the random variables, Xi, from a geometric distribution is a sufficient statistic. A sufficient statistic is a statistic that includes all the information in the sample that is relevant to the parameter being estimated.

Here are the steps to show this:

1. First, we need to write down the joint probability mass function (pmf) of the sample. Since the observations are independent, the joint pmf is the product of the individual pmfs. So, we have:

p(X1, X2, ..., Xn; π) = Π_{i=1}^{n} p(Xi; π)
= Π_{i=1}^{n} (1 - π)^(Xi - 1) * π
= π^n * (1 - π)^(Σ_{i=1}^{n} (Xi - 1))
= π^n * (1 - π)^(Σ_{i=1}^{n} Xi - n)

2. Now, we can see that the joint pmf can be expressed as a function of π and T(X), where T(X) = Σ_{i=1}^{n} Xi, and does not depend on the individual Xi's. This is the definition of a sufficient statistic.

Therefore, the sum of the Xi's, Σ_{i=1}^{n} Xi, is a sufficient statistic for the parameter π in a geometric distribution.

Question

The problem is asking to show that the sum of the random variables, Xi, from a geometric distribution is a sufficient statistic. A sufficient statistic is a statistic that includes all the information in the sample that is relevant to the parameter being estimated.

Here are the steps to show this:

1. First, we need to write down the joint probability mass function (pmf) of the sample. Since the observations are independent, the joint pmf is the product of the individual pmfs. So, we have:

p(X1, X2, ..., Xn; π) = Π_{i=1}^{n} p(Xi; π) 
                         = Π_{i=1}^{n} (1 - π)^(Xi - 1) * π
                         = π^n * (1 - π)^(Σ_{i=1}^{n} (Xi - 1))
                         = π^n * (1 - π)^(Σ_{i=1}^{n} Xi - n)

2. Now, we can see that the joint pmf can be expressed as a function of π and T(X), where T(X) = Σ_{i=1}^{n} Xi, and does not depend on the individual Xi's. This is the definition of a sufficient statistic.

Therefore, the sum of the Xi's, Σ_{i=1}^{n} Xi, is a sufficient statistic for the parameter π in a geometric distribution.

Knowee AI · Accepted Answer

The problem is asking to show that the sum of the random variables, Xi, from a geometric distribution is a sufficient statistic. A sufficient statistic is a statistic that includes all the information in the sample that is relevant to the parameter being estimated.

Here are the steps to show this:

1. First, we need to write down the joint probability mass function (pmf) of the sample. Since the observations are independent, the joint pmf is the product of the individual pmfs. So, we have:

p(X1, X2, ..., Xn; π) = Π_{i=1}^{n} p(Xi; π) 
                         = Π_{i=1}^{n} (1 - π)^(Xi - 1) * π
                         = π^n * (1 - π)^(Σ_{i=1}^{n} (Xi - 1))
                         = π^n * (1 - π)^(Σ_{i=1}^{n} Xi - n)

2. Now, we can see that the joint pmf can be expressed as a function of π and T(X), where T(X) = Σ_{i=1}^{n} Xi, and does not depend on the individual Xi's. This is the definition of a sufficient statistic.

Therefore, the sum of the Xi's, Σ_{i=1}^{n} Xi, is a sufficient statistic for the parameter π in a geometric distribution.

Let X = {X1, X2, . . . , Xn} be a random sample of size n from the geometricdistribution with probability mass function:pX (x; π) ={(1 − π)x−1π for x = 1, 2, . . .0 otherwisewhere π is the probability of success. Show that n∑i=1Xi is a suﬃcient statisti

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