Question

Here \(Q_{1}\) and \(Q_{2}\) are quadratic forms in observations of a random sample from \(N(0,1) .\) If \(Q_{1}\) and \(Q_{2}\) are independent and if \(Q_{1}+Q_{2}\) has a chi-square distribution, prove that \(Q_{1}\) and \(Q_{2}\) are chi-square variables.

Short answer

Using properties of chi-square distributions, it can be proven that both \(Q_{1}\) and \(Q_{2}\) are chi-square distributed.

Step-by-step solution

Understand the Assumptions

According to the problem, \(Q_{1}\) and \(Q_{2}\) are independent, and \(Q_{1}+Q_{2}\) is chi-square distributed. These are given conditions and do not need to be proven.

Apply property of chi-square distribution

The property of the chi-square distribution state that the sum of independent chi-square distributed random variables results in another chi-square distribution. Therefore, if \(Q_{1}+Q_{2}\) is chi-square distributed, and \(Q_{1}\) and \(Q_{2}\) are independent, then \(Q_{1}\) and \(Q_{2}\) must each be chi-square distributed by the property of additive chi-square distributions.

Conclusion

Based on the assumptions, the additive property of the chi-square distribution and the independence of \(Q_{1}\) and \(Q_{2}\), it can be concluded that \(Q_{1}\) and \(Q_{2}\) must be chi-square distributed.

Key concepts

Quadratic forms

Quadratic forms represent expressions where variables are squared and sometimes multiplied by each other. In matrix terms, if you have a vector \( \mathbf{x} = (x_1, x_2, \ldots, x_n) \), a quadratic form in \( \mathbf{x} \) can be expressed as \( \mathbf{x}^T A \mathbf{x} \), where \( A \) is an \( n \times n \) square matrix.

The quadratic forms can also be tied to probability, particularly in statistics when examining certain distributions such as the chi-square distribution. When considering random variables sampled from a normal distribution, these forms play a key role.

In the context of the given exercise, \( Q_1 \) and \( Q_2 \) are quadratic forms derived from observations of a sample from a normal distribution. The nature of these quadratic forms and their resulting distributions provide insights into the characteristics and properties of the variables involved.

Independent random variables

Independence between random variables is a fundamental concept in probability and statistics. When two random variables \( X \) and \( Y \) are independent, the occurrence of any event related to \( X \) does not give us any information about \( Y \) and vice versa.

Mathematically, \( X \) and \( Y \) are independent if the joint probability distribution satisfies \( P(X \cap Y) = P(X) \cdot P(Y) \), meaning that the joint distribution is the product of their marginal distributions. This concept simplifies many analyses because the behavior of independent random variables can be studied separately.

• In the provided problem, the independence of \( Q_1 \) and \( Q_2 \) helps us directly apply known properties such as the summing of distributions.
• This property ensures that manipulations on one quadratic form do not affect the other.

Understanding independence is crucial when dealing with complex probability models and confirms why certain distributions, such as the chi-square, maintain their form under summation.

Properties of chi-square distribution

The chi-square distribution is a fundamental probability distribution often encountered in statistical practices. It arises when trying to determine how a sample variance compares to a known population variance. This distribution is crucial for goodness-of-fit tests and tests of independence in contingency tables.

One of its key properties is additive nature: if \( X_1 \) and \( X_2 \) are independent chi-square random variables with \( k_1 \) and \( k_2 \) degrees of freedom respectively, then \( X_1 + X_2 \) is also chi-square distributed with \( k_1 + k_2 \) degrees of freedom. This is the property used in the original problem to conclude that \( Q_1 \) and \( Q_2 \) must also be independently chi-square distributed if their sum is a chi-square variable.

• The distribution is skewed to the right, meaning it is not symmetric like the normal distribution, especially for fewer degrees of freedom.
• As the number of degrees of freedom increases, the chi-square distribution approaches a normal distribution.
• It is only defined for non-negative numbers, reflecting its creation from squared outcomes.

These properties allow the chi-square distribution to be widely used in analytical statistics for hypothesis testing and variance analysis.