Question

Let \(X_{1}\) and \(X_{2}\) be two independent random variables. Let \(X_{1}\) and \(Y=\) \(X_{1}+X_{2}\) be \(\chi^{2}\left(r_{1}, \theta_{1}\right)\) and \(\chi^{2}(r, \theta)\), respectively. Here \(r_{1}

Short answer

Following the analysis and solution steps, given that \(X_{1}\) ~ \(\chi^{2}(r_{1}, \theta_{1})\) and \(Y = X_{1} + X_{2}\) ~ \(\chi^{2}(r, \theta)\) with \(r_{1} < r\) and \(\theta_{1} \leq \theta\), \(X_{2}\) can be shown to follow a \(\chi^{2}(r-r_{1}, \theta-\theta_{1})\) distribution.

Step-by-step solution

Understanding the chi-square distribution

Firstly, it's important to know the properties of a chi-square distribution. If a random variable follows a chi-square distribution, this variable is constructed as the sum of the squares of independent standard normal random variables. In this case, \(r_{1}\) and \(r\) represent the degrees of freedom, also \(\theta_{1}\) and \(\theta\) are the scale parameters. We know that, if \(X_{1}\) ~ \(\chi^{2}(r_{1}, \theta_{1})\) and \(Y\) ~ \(\chi^{2}(r, \theta)\), then \(Y\) is actually \(X_{1} + X_{2}\), and \(X_{1}\) and \(X_{2}\) are independent.

Substitute \(Y = X_{1} + X_{2}\) into the distribution of \(Y\)

According to the given condition, \(Y = X_{1} + X_{2}\). Thus, we can substitute this into the distribution of \(Y\). As \(X_{1}\) and \(X_{2}\) are independent, we have \(Y\) ~ \(\chi^{2}(r_{1}+k, \theta_{1}+l)\), where \(k\) and \(l\) are parameters of \(X_{2}\) that we wish to find.

Compare parameters

We can now equate parameters of \(Y\). The degrees of freedom \(r\) for \(Y\) is equal to the sum of the degrees of freedom for \(X_{1}\) and \(X_{2}\), i.e \(r_{1} + k = r\). Similarly, the scale parameter \(\theta\) for \(Y\) is equal to the sum of the scale parameters for \(X_{1}\) and \(X_{2}\), i.e \(\theta_{1} + l = \theta\).

Calculate the parameters for \(X_{2}\)

By solving the equations, we find that \(k = r - r_{1}\) and \(l = \theta - \theta_{1}\). So, if we let \(X_{2}\) ~ \(\chi^{2}(k, l)\), it will precisely indicate \(X_{2}\) ~ \(\chi^{2}(r-r_{1}, \theta-\theta_{1})\), as required by the problem.

Key concepts

Independent Random Variables

Independent random variables are a foundational concept in statistics, particularly when analyzing the behavior and relationship between different variables. In essence, two random variables are considered independent if the occurrence of one does not affect the probability of occurrence of the other. This means that knowing the outcome of one gives no information about the outcome of the other.

For instance, in the given exercise, we have two random variables, \(X_{1}\) and \(X_{2}\), that are defined as independent. This is a critical condition as it allows for the sum of these variables to be treated in a specific way when working with distributions, particularly the chi-square distribution in this scenario. Understanding this independence is key to correctly applying the properties that define their combined distribution.

Degrees of Freedom

The concept of degrees of freedom is often mentioned in statistical analyses, especially in relation to distributions such as the chi-square distribution. Degrees of freedom refer to the number of values in a calculation that are free to vary. To put it simply, they are a measure of how much certain values can change in the statistics before affecting other variables or the outcome.

In our exercise, degrees of freedom are represented by \(r_{1}\) and \(r\) for the distributions of \(X_{1}\) and \(Y\) respectively. The significance of degrees of freedom becomes evident when we examine distributions resulting from the combination of independent random variables. The degrees of freedom for the combined variable \(Y\), which symbolizes \(X_{1} + X_{2}\), is the sum of the individual degrees of freedom from \(X_{1}\) and \(X_{2}\). This sum reflects the total 'space' in which our distribution can vary.

Scale Parameter

When it comes to the chi-square distribution, the scale parameter is an important aspect that affects the spread of the distribution. In the broad spectrum of probability distributions, the scale parameter modifies the variance without changing the nature of the distribution. For the chi-square distribution, the scale parameter, represented by \(\theta\), essentially scales the standard deviation, altering the width of the distribution curve without affecting its shape.

The exercise poses an interesting scenario where the scale parameter is used to identify the chi-square distribution of \(X_{2}\), after acknowledging the given parameters \(\theta_1\) for \(X_{1}\) and \(\theta\) for \(Y\). What this implies in practice is that the scale parameters of the independent random variables, when summed, should match the scale parameter of the resulting combination. This serves to maintain the inherent properties of the chi-square distribution after the addition of two such variables.