Question

. Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a Poisson distribution with mean \(\mu .\) Thus \(Y=\sum_{i=1}^{n} X_{i}\) has a Poisson distribution with mean \(n \mu .\) Moreover, \(\bar{X}=Y / n\) is approximately \(N(\mu, \mu / n)\) for large \(n .\) Show that \(u(Y / n)=\sqrt{Y / n}\) is a function of \(Y / n\) whose variance is essentially free of \(\mu .\)

Short answer

It can be proved that the variance of \(u(Y/n) = \sqrt{Y/n}\) is free of \( \mu\) as \(n\) becomes very large, making \( \mu / n\) approach zero regardless of the value of \( \mu\).

Step-by-step solution

Define Variables

Let's first define the given variables. We have a random sample, \(X_{1}, X_{2}, \ldots, X_{n}\), from a Poisson distribution with an average (\( \mu\)). The sample sum \(Y=\sum_{i=1}^{n} X_{i}\) also follows a Poisson distribution but with a mean of \(n \mu \). Then, we have \(\overline{X} = Y/n\), which is approximated to be \(N(\mu, \mu / n)\), for large \(n\). We're asked to show that \(u(Y/n) = \sqrt{Y/n}\) is a function of \( Y/n\) whose variance is free from \( \mu\).

Define the Function u(Y/n)

We need to define \(u(Y/n) = \sqrt{Y/n}\). By squaring both sides of this equation, we get \(u^2(Y/n) = Y/n\), or simply \(u^2 = \overline{X}\).

Calculate the Variance

Next, we calculate the variation of the function u which is \(Var(u^2(Y/n)) = Var(\overline{X})\). Since it's given that \(\overline{X}\) approximates \(N(\mu, \mu / n)\) for a large \(n\), then \(Var(\overline{X}) = \mu / n\).

Show the Variance is Essentially Free of \(\mu\)

Finally, it is clear that the variance, \(Var(u^2(Y/n)) = \mu / n\) is essentially free of \( \mu\). This is because as \(n\) (the size of the sample) becomes very large, the \( \mu / n\) will approach zero, regardless of \( \mu\). In effect, \( \mu\) will have minimal to no influence on the variance for large \(n\).

Key concepts

Understanding a Random Sample

A random sample refers to a set of observations selected from a larger population with each member having an equal chance of being chosen.This selection method is crucial in statistical analysis, ensuring that results are unbiased and representative of the entire population.For example, in our Poisson distribution problem, each observation, denoted as \(X_1, X_2, \ldots, X_n\), is independently selected from the population.
When we say these observations come from a Poisson distribution, it means each \(X_i\) follows a specific probability distribution characterized by the average number of events \(\mu\) within a fixed interval of time or space.To summarize:

• Every element in the sample has an equal opportunity to be part of the sample.
• This helps create results that accurately reflect the characteristics of the whole group.
• Random sampling minimizes bias and errors in statistical conclusions.

What is Variance?

Variance, in statistics, measures the degree to which numbers in a data set are spread out.In simpler terms, it tells us how much the numbers differ from the mean (average) of the set.It's calculated by taking the average of the squared differences between each value and the mean.
In our problem context, we deal with the variance of a random variable \(\overline{X}\).For a large sample, \(\overline{X}\) approximates a normal distribution with a mean \(\mu\) and variance \(\mu/n\).This variance signifies the distribution of \(\overline{X}\) around its mean.To put it visually:

• Variance provides an idea of the data's spread in relation to the mean.
• It is crucial for understanding how the data fluctuates.
• In the Poisson distribution, variance \(\mu/n\) shows how much \(\overline{X}\) can deviate from \(\mu\).

Explaining Normal Approximation

Normal approximation is a technique used to approximate a probability distribution by a normal distribution.In essence, for large samples, many distributions tend to resemble a normal distribution due to the Central Limit Theorem.
In our discussion, the variable \(\overline{X}\) is approximately normal when the sample size \(n\) is large enough.This means the distribution of \(\overline{X}\) can be transformed into standard normal form with mean \(\mu\) and variance \(\mu/n\).Here are the key takeaways:

• Normal approximation simplifies complex distributions for ease of calculation.
• It is especially useful when dealing with large datasets or samples.
• Helps in applying statistical tests and making inferences about the data.

Diving into the Concept of Mean

The mean is a fundamental statistical concept representing the average value in a data set.It is calculated by summing up all the numbers and dividing by the count of numbers.
In the case of a Poisson distribution, the mean \(\mu\) is not just an average; it describes the expected number of occurrences in a given unit of space or time.When dealing with our problem's variable \(\overline{X}\), the mean remains \(\mu\) while the variance is adjusted based on sample size \(n\).Here are a few things to keep in mind:

• Mean provides a central value for a data set and influences its overall distribution.
• In a Poisson distribution, the mean also indicates the event rate.
• In statistical analysis of large samples, knowing the mean helps in understanding the distribution.