TITLE: Compute the moment generating function of $Y = X_1X_2 + X_1X_3 + X_2X_3$
QUESTION [5 upvotes]: Suppose $X_1, X_2,$ and $X_3$ are independent and $N(0, 1)$-distributed. Compute the moment generating function of $Y = X_1X_2 + X_1X_3 + X_2X_3$.

I know that any $X_iX_j$ with $i \not =j $ is a joint normal with variables $(x_i,x_j)$
I also know the formula of the moment generating function of a normal
distribution.
Furthermore, I know that if $Y_1,…, Y_n$ are independent $N(0,1)$, that is, $Y = (Y_1,…,Y_n )´$ are $N(0,I)$ by definition, the moment generating  function of Y is given by $$e^{\frac{1}{2}\mathbf t' \mathbf t}$$

I thought about using the pdf's and the definition of a moment generating function but it proved to be a really tedious process jacked of multiple integrations.
Does anyone know how to easily solve this problem with some relatively simple lines? (Especially using the multivariate normal properties and matrices)

REPLY [4 votes]: Define
$$A=\begin{bmatrix}
0&1&1\\
1&0&1\\
1&1&0
\end{bmatrix},\quad X=\begin{bmatrix}
X_1\\
X_2\\
X_3
\end{bmatrix}$$
the other choices of $A$ would give the correct $Y$, but I need it to be symmetric later. Then
$$
Y=\frac{1}{2}X^TAX
$$
The pdf for 3d standard normal distribution
$$
p(x)=(2\pi)^{-3/2}\exp\left[-\frac{1}{2}x^Tx\right]
$$
The moment generating function for y is then
$$
\mathbb{E}(e^{\lambda y})=\int d^3x(2\pi)^{-3/2}\exp\left[\frac{\lambda}{2}x^TAx\right]\exp\left[-\frac{1}{2}x^Tx\right]=(2\pi)^{-3/2}\int d^3x\exp\left[-\frac{1}{2}x^T(I-\lambda A)x\right]
$$
The above integral is solved by the following for a real symmetric matrix $M$
$$
\int d^3x\exp\left[-\frac{1}{2}x^TMx\right]=\sqrt{\frac{(2\pi)^3}{\det M}}
$$
Inserting we find
$$
\mathbb{E}(e^{\lambda y})=\frac{1}{\sqrt{\det(I-\lambda A)}}=\frac{1}{\sqrt{-2\lambda^3-3\lambda^2+1}}
$$