Question

Let \(X \sim \chi_{v_{1}}^{2}\) and \(Y \sim \chi_{v_{2}}^{2} .\) Prove that \(Z=\frac{X / v_{1}}{Y / v_{2}} \sim F\left(v_{1}, v_{2}\right)\).

Short answer

The random variable \(Z=\frac{X / v_{1}}{Y / v_{2}}\) can be recognised as a ratio of two independent chi-squared distributed variables, both divided by their degrees of freedom, meaning by the definition of the F-distribution, that \(Z\) follows an F-distribution \(F\left(v_{1}, v_{2}\right)\).

Step-by-step solution

Define The Random Variables

We have two random variables, \(X\) and \(Y\), which are chi-square distributed with \(v_1\) and \(v_2\) degrees of freedom respectively. \(Z\) is a random variable defined as the ratio \(Z=\frac{X / v_{1}}{Y / v_{2}}\). We have to show that \(Z\) is F-distributed.

Define F-Distribution

The F-distribution is the ratio of two chi-square distributions. Specifically, if \(U \sim \chi_{d1}^2\) and \(V \sim \chi_{d2}^2\), and \(U\) and \(V\) are independent, then the random variable \(F = \frac{U / d1}{V / d2}\) follows an F-distribution with parameters \(d1\) and \(d2\). In our case, the parameters are \(v1\) and \(v2\).

Demonstrate That Z is F-Distributed

By definition, the random variable \(Z=\frac{X / v_{1}}{Y / v_{2}}\) falls under the given formulation for an F-distributed variable since \(X\) and \(Y\) are independently chi-square distributed. Therefore, \(Z\) follows an F-distribution with parameters \(v1\) and \(v2\).

Key concepts

Chi-Square Distribution

The chi-square distribution is a statistical distribution commonly used to analyze the variance within a dataset. It is especially useful in hypothesis testing and is the foundation for the widely used chi-square test of independence. The chi-square distribution arises naturally as the sum of the squares of independent standard normal random variables. For each chi-square variable, there is an associated degrees of freedom ((v)), which corresponds to the number of independent pieces of information involved. In the case of the exercise, random variables \(X\) and \(Y\) each have their own chi-square distributions with degrees of freedom \(v_1\) and \(v_2\) respectively.

When calculating with chi-square distributions, it's important to understand that larger degrees of freedom generally lead to a distribution that closely resembles the normal distribution. However, with few degrees of freedom, the chi-square distribution is skewed to the right, indicating that extreme values are more likely. Understanding this property can help students properly interpret the results of statistical tests that use this distribution.

Degrees of Freedom

Degrees of freedom ((df)) are a crucial concept in statistical analysis, representing the number of independent values that can vary in an analysis without breaking any constraints. They are essentially the number of values in the final calculation of a statistic that are free to vary. The degrees of freedom are directly linked to the number of observations in your data and the number of parameters you need to estimate for a given model.

In the context of the exercise, \(v_1\) and \(v_2\) represent the degrees of freedom for the chi-square distributions of \(X\) and \(Y\), respectively. It's important because the shape of the chi-square distribution changes with different degrees of freedom. Hence, students must grasp how degrees of freedom influence their statistical analysis, as this will affect the accuracy and interpretability of their results.

Statistical Distributions

Statistical distributions provide a framework for understanding and predicting the frequencies of different outcomes within a dataset. They offer a visual representation of the data, conveying information about the variability, central tendency, and likelihood of different outcomes. Common distributions include the normal, binomial, Poisson, and, as highlighted in this exercise, the chi-square and F-distributions.

Understanding various statistical distributions enables students to select appropriate models for their data analysis tasks. For instance, when a dataset is compared against a particular distribution, it implies assumptions about the data's randomness, variability, and form. Acknowledging these distributions, their shapes, and their parameters (like mean and variance) is crucial for accurate statistical inference and for understanding exercises like the one given, where a variable follows an F-distribution.

Probability Theory

Probability theory is the branch of mathematics concerned with the analysis of random phenomena. It provides the foundation for statistical distributions, as it allows us to assign a measure to the likelihood of various outcomes. This theory underpins all of statistical analysis and aids in the understanding of concepts such as the chi-square and F-distributions.

In probability theory, the idea of independent events is key to many statistical concepts. In our exercise, the independence of chi-square distributed random variables \(X\) and \(Y\) is essential to establish that their ratio follows an F-distribution. Furthermore, probability theory elucidates how the degrees of freedom affect the shape and spread of distributions. For students, developing a solid understanding of probability calculation methods and the logic behind them is integral to employing statistical methods in data analysis properly.