Question

A binomial variable \(R\) has mean \(m \pi\) and variance \(m \pi(1-\pi) .\) Find the variance function of \(Y=R / m\), and hence obtain the variance-stabilizing transform for \(R\).

Short answer

The variance-stabilizing transform for \( R \) is \( h(Y) = \sqrt{m} \cdot \arcsin(\sqrt{Y}) \).

Step-by-step solution

Understand the Problem

We are given a binomial variable \( R \) with mean \( m\pi \) and variance \( m\pi(1-\pi) \). We need to find the variance function of \( Y = \frac{R}{m} \) and use it to derive the variance-stabilizing transformation for \( R \).

Define Random Variable Y

The random variable \( Y = \frac{R}{m} \) can be thought of as the proportion of successes in the binomial experiment. \( Y \) has the mean \( \frac{m\pi}{m} = \pi \).

Calculate the Variance of Y

Using the definition of variance, the variance of \( Y \) is calculated as follows:\[ \text{Var}(Y) = \text{Var}\left(\frac{R}{m}\right) = \frac{1}{m^2} \text{Var}(R) = \frac{1}{m^2} \cdot m\pi(1-\pi) = \frac{\pi(1-\pi)}{m} \]

Find the Variance Function of Y

From the above calculation, we deduce that the variance function of \( Y \) is \( \text{Var}(Y) = \frac{\pi(1-\pi)}{m} \).

Determine the Variance-Stabilizing Transform

The variance-stabilizing transformation seeks to make the variance of the transformed variable approximately constant. We use the transform:\[ h(Y) = \int \frac{1}{\sqrt{\text{Var}(Y)}} dY \]In this case, we need to integrate:\[ \int \frac{1}{\sqrt{\frac{\pi(1-\pi)}{m}}} dY = \sqrt{m} \int \frac{1}{\sqrt{\pi(1-\pi)}} dY \]

Simplify the Integral

To further proceed, note that \( \pi = Y \). Hence, the integral becomes:\[ \sqrt{m} \int \frac{1}{\sqrt{Y(1-Y)}} dY \]This is a known result and leads to the transform:\[ h(Y) = \sqrt{m} \cdot \arcsin(\sqrt{Y}) \]

Key concepts

Understanding the Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It's like flipping a coin multiple times and counting how many times it lands on heads. Given a binomial random variable, such as \( R \), it has the mean \( m \pi \) and variance \( m \pi (1 - \pi) \), where \( m \) represents the number of trials and \( \pi \) is the probability of success in a single trial. These formulas are derived from the properties of independent trials: the mean is the expected number of successes, and the variance measures the spread of the distribution around the mean.

Key features of the binomial distribution include:

• A fixed number of trials \( m \)
• Each trial results in success or failure
• The probability \( \pi \) of success is the same on every trial
• Trials are independent of each other

Understanding these components is essential for solving problems involving the binomial distribution.

Exploring the Variance Function

The variance function is an important concept because it describes how the variance of a probability distribution scales with the parameters of the distribution. In our exercise, we investigated the variance function of the variable \( Y = \frac{R}{m} \).

Here, \( Y \) represents the proportion of successes, and its variance was derived as \( \text{Var}(Y) = \frac{\pi(1-\pi)}{m} \). This function shows that the variance is inversely proportional to \( m \), meaning that as the number of trials increases, the variance of our proportion \( Y \) decreases, contributing to more certainty about \( Y \).

This pattern is typical of binomial distributions, where increased sample sizes lead to reduced variance, consequently stabilizing the expectation around the mean.

Transformation of Variables and Variance-Stabilizing Transform

Transforming variables is a crucial technique in statistics, especially when dealing with non-constant variance. The goal is to create a transformation that stabilizes this variance across different values of the variable. In our exercise, we aim to find a variance-stabilizing transformation for the binomial variable \( R \).

To achieve this, we use the transformation \( h(Y) = \int \frac{1}{\sqrt{\text{Var}(Y)}} dY \). By integrating, we find a transformation \( h(Y) \) that approximately equalizes the variance across different values of the outcome \( Y \).

In the example, this results in \( h(Y) = \sqrt{m} \cdot \arcsin(\sqrt{Y}) \), revealing that the arcsin function is particularly useful for this transformation. By stabilizing variance, one can make more reliable statistical inferences from the sample data. Such transformations are widely used in practice, especially in scenarios where homogeneity of variance is assumed or required for valid analysis.