Given a multivariate normal random $n\times 1$ vector $X \sim N(\mu,\Sigma)$, what is the expectation $$\mathbb{E}[exp(X^TAX+b^TX)]$$ where $A$ is a $n\times n$ matrix and $b$ is a n-dimensional vector? I know that $\mathbb{E}[X^TAX] = \mathbb{E}[X^T]A\mathbb{E}[X]+tr(A\Sigma)$, but how should I solve this problem?


$$ X\sim N(\mu,\Sigma)\quad\Rightarrow\quad \text{pdf}=f_X(X)=\frac{1}{\sqrt{(2\pi)^n|\Sigma|}}\exp(-\frac{1}{2}(X-\mu)^T\Sigma^{-1}(X-\mu)) $$ Therefore: $$ \begin{split} E[\exp(X^TAX+b^TX)]&=\int_{\mathbb R^n}\frac{1}{\sqrt{(2\pi)^n|\Sigma|}}\exp(-\frac{1}{2}(X-\mu)^T\Sigma^{-1}(X-\mu)+X^TAX+b^TX)dX\\ &=\int_{\mathbb R^n}\frac{1}{\sqrt{(2\pi)^n|\Sigma|}}\exp(-\frac{1}{2}(X-\Delta)^T\Sigma'^{-1}(X-\Delta)+\Lambda)dX \end{split} $$ where $$ \Sigma'=(\Sigma^{-1}-2A)^{-1}, \Delta=\Sigma'(b+\Sigma^{-1}\mu), \Lambda=\frac{1}{2}(\Delta^T\Sigma'^{-1}\Delta-\mu^T\Sigma^{-1}\mu). $$ Therefore: $$ \begin{split} E[\exp(X^TAX+b^TX)]&=\sqrt{\frac{|\Sigma'|}{|\Sigma|}}\exp(\Lambda)\int_{\mathbb R^n}\frac{1}{\sqrt{(2\pi)^n|\Sigma'|}}\exp(-\frac{1}{2}(X-\Delta)^T\Sigma'^{-1}(X-\Delta))dX\\ &=\sqrt{\frac{|\Sigma'|}{|\Sigma|}}\exp(\Lambda) \end{split} $$

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  • $\begingroup$ Thanks for replying. I have two questions to ask you if you don't mind: 1. What is the explanation for the first line of equation you have written after "Therefore:"? I found a mention of a similar problem at this link goo.gl/fNqvoA and I can see that your answer can be simplified to the form given in this link, but I am having problem understanding the first step of your solution itself. 2. Can you suggest me a book to study this topic in detail so that I can solve the problems on my own? $\endgroup$ – AbbyJ Aug 24 '14 at 14:59
  • $\begingroup$ @abbyj The link you gave refer to a completely different problem. Do you know what a closed form means? $\endgroup$ – Troy Woo Aug 24 '14 at 15:10
  • $\begingroup$ yes and I guess you mean that the closed form expression here shouldn't contain any integrals over infinite space, right? $\endgroup$ – AbbyJ Aug 24 '14 at 15:16
  • $\begingroup$ @abbyj multivariate normal distribution is much like the single variable one if you use diagonalization. What I did is basically completion of squares, which is high school math. The rest is utilizing the fact that probability density function integrates to $1$ over the entire domain $\mathbb R^n$. There are plenty of book on this subject, I didn't read a single one so I cant recommend you any reference. $\endgroup$ – Troy Woo Aug 24 '14 at 15:23
  • $\begingroup$ Got your explanation. Thanks for helping. $\endgroup$ – AbbyJ Aug 24 '14 at 15:37

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