I am trying to solve a problem in probability theory which is as follows:
Problem:
Suppose $X_1,...,X_n$ are i.i.d. with distribution $\mathcal{N}(0,1)$ and $a_1,...,a_n\in \mathbb R$ are such that $\sum\limits_{i} a_i^2=1$. Let $Y=\sum\limits_{i}a_iX_i$. Then show that $Y$ is independent of $\sum\limits_{i} X_i^2-Y^2$.
I have managed to show that $Y$ follows the standard normal distribution. But I am unable to show why the two random variables are independent of each other. Can someone help me?
Choose an orthogonal matrix $A$ whose first row is $(a_1~\ldots~a_n)$. It is possible since this row is a unit vector. Define $Y_1,\ldots,Y_n$ by $$(Y_1~\ldots~Y_n)^\top := A(X_1~\ldots~X_n)^\top.$$ Then $Y_1$ is the random variable $Y$ you considered.
Since $X_1,...,X_n$ are i.i.d. with distribution $\mathcal{N}(0,1)$, $(X_1~\ldots~X_n)^\top$ is Gaussian with distribution $\mathcal{N}(0,I_n)$, so $(X_1~\ldots~X_n)^\top$ is Gaussian with distribution $\mathcal{N}(0,AI_nA^\top) = \mathcal{N}(0,I_n)$. Hence $Y_1,...,Y_n$ are i.i.d. with distribution $\mathcal{N}(0,1)$. In particular, $Y_1$ is independent of $Y_2^2 + \cdots + Y_n^2 = X_1^2 + \cdots + X_n^2 - Y_1^2$.