Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample of size \(n\) from a distribution which is \(N\left(0, \sigma^{2}\right)\). Prove that \(\sum_{1}^{n} X_{i}^{2}\) and every quadratic form, which is nonidentically zero in \(X_{1}, X_{2}, \ldots, X_{n}\), are dependent.

Short answer

The given quadratic form and sum of squares of \(X_i\)'s are not independent because the quadratic form contains the sum of squares of the variables.

Step-by-step solution

Understanding Variables

Consider the quadratic form \( Q = a_1X_1^2 + a_2X_2^2 + ... + a_nX_n^2 \), where \(a_i\) are constant scalars and \(X_i\) are the random variables. Here, \(X_i\) are the random sample of size \(n\) and they are independent (by definition of a random sample) and identically distributed as \(N\left(0, \sigma^{2}\right)\).

Understanding Dependent and Independent Variables

The variables \(X_i\) are said to be linearly dependent if scalar multiples of them sum to zero. For \(X_i\), they are linearly dependent if there exist coefficients \(a_i\) not all of which are zero such that \(a_1X_1 + a_2X_2 + ... + a_nX_n = 0\). As \(X_i\) are independent (definition of being a random sample), the sum \( S = X_1 + X_2 + ... + X_n\) is a linear combination of \(X_i\).

Evaluate the Quadratic form and the Sum of Squares

For quadratic form \(Q\), we should note that \(Q\) includes squares of \(X_i\). It's clear that the form \(S = X_1 + X_2 + ... + X_n\) is contained in \(Q\) when considering their square components. This implies that the quadratic form \(Q\) and the sum of squares of \(X\)'s are not independent.