# How to derive the sampling distribution of some statistics

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In summary: However, from the given information, it is possible to derive formulas for the sampling distribution of the coefficient of variation and skewness with a sample size ##n##.
TL;DR Summary
Given a geometric Erlang distribution, how can I derive the sampling distribution of the coefficient of variation and the skewness for a sample of size $n$.
Assume that ##T## has an Erlang distribution:
$$\displaystyle f \left(t \, | \, k \right)=\frac{\lambda ^{k }~t ^{k -1}~e^{-\lambda ~t }}{\left(k -1\right)!}$$
and ##K## has a geometric distribution
$$\displaystyle P \left( K=k \right) \, = \, \left( 1-p \right) ^{k-1}p$$
Then the compound distribution has the following form.
$$\displaystyle g \left(t \right)= \sum _{k=1}^{\infty} f \left(t \, | \, k \right)~P \left(K =k \right)=\frac{\lambda ~p }{e^{\lambda ~t ~p }}$$
with expectation:
$$\displaystyle \mu_{{1}}\, = \,{\frac {1}{\lambda\,p}}$$
variance:
$$\displaystyle \mu_{{2}}\, = \,{\frac {1}{{\lambda}^{2}{p}^{2}}}$$
and third central moment:
$$\displaystyle \mu_{{3}}\, = \, {\frac {2}{{\lambda}^{3}{p}^{3}}}$$
The coefficient of variation ##c_v## is given by:
$$\displaystyle {\it c_v}\, = \,{\frac { \sqrt{\mu_{{2}}}}{\mu_{{1}}}}=1$$
and the skewness ##\tilde{\mu}_3## by:
$$\displaystyle {\it \tilde{\mu}_3}\, = \,{\frac {\mu_{{3}}}{{\mu_{{2}}}^{3/2}}}=2$$
Is it possible to derive a formula for the sampling distribution of the coefficient of variation and the skewness with a sample size ##n##?

Last edited:
mathman said:

I am now aware that ##g(t)## is in fact an exponential distribution with rate parameter ##\lambda \, p##. But the Wikipedia site on the exponential distribution makes no mention of sampling distributions.

Section on statistical parameters? I am not sure what you want.

## 1. What is a sampling distribution?

A sampling distribution is a theoretical probability distribution that represents the possible values of a statistic, such as the mean or standard deviation, that could be obtained from different samples of a population. It is derived by repeatedly sampling from a population and calculating the statistic for each sample.

## 2. Why is it important to derive the sampling distribution of a statistic?

Deriving the sampling distribution allows us to make inferences about a population using sample data. It helps us understand the variability of a statistic and how likely it is to differ from the true population parameter. This is essential for making accurate conclusions and predictions in scientific research.

## 3. How do you derive the sampling distribution of a statistic?

The sampling distribution of a statistic can be derived using mathematical formulas or through simulation methods. For example, the sampling distribution of the mean can be derived by calculating the mean of all possible samples of a given size from a population and plotting the results on a graph.

## 4. What factors influence the shape of a sampling distribution?

The shape of a sampling distribution is influenced by the sample size, the shape of the population distribution, and the variability of the population. As the sample size increases, the sampling distribution tends to become more normal. A population with a non-normal distribution or high variability may result in a non-normal sampling distribution.

## 5. Can the sampling distribution of a statistic be used to make predictions about individual values?

No, the sampling distribution of a statistic represents the distribution of values that the statistic could take on, not the actual values themselves. It is used to make inferences about a population, not individual values. However, it can be used to calculate the probability of obtaining a certain value or range of values for the statistic.

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