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.