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I'm trying to solve the following problem

Use Monte-Carlo methods to find a $95 \%$ confidence interval for the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp\left( \frac{1}{2}\left[x^2 - \frac{x(y-1)}{10} +(y-1)^2\right] \right ) dx dy$$

Now, my first thoughts were that this function looks similar to a bivariate normal density. For example, considering a bivariate normal distribution with parameters $\mu_1 = 0, \mu_2 = \sigma_1 = \sigma_2 = 1$ and $\rho = \frac{1}{20}$ we would have the density

$$f(x,y) = \frac{1}{2 \pi \sqrt{399/400}} \ \exp\left(-\frac{400}{399} \ g(x,y) \right)$$ where $g(x)$ is the function being integrated, that is,

$$ g(x)= \exp\left( \frac{1}{2}\left[x^2 - \frac{x(y-1)}{10} +(y-1)^2\right] \right ). $$

I have tried to express the integral as some kind of expected value under the bivariate normal distribution, so as to simulate values under that distribution, but somehow I always end up getting stuck. If anyone has any idea on how to proceed, I would appreciate it very much.

Thanks!

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    $\begingroup$ The normal distribution has the shape $\exp(\mathbf{-} x^2)$, not $\exp(x^2)$. $\endgroup$ Commented Oct 10, 2020 at 5:04
  • $\begingroup$ Agreed, that's why I said similar. I was still hoping there might be a way to write the expression as an expectation of the bivariate normal density. $\endgroup$
    – Bergson
    Commented Oct 10, 2020 at 5:44

1 Answer 1

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Unless there is a typo in the text of the exercise (and this is quite likely!), Monte Carlo methods are superfluous for computing this integral since the integrand diverges at $\pm\infty$. Hence $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp\left( \frac{1}{2}\left[x^2 - \frac{x(y-1)}{10} +(y-1)^2\right] \right )\, \text{d}x\,\text{d}y=+\infty$$

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  • $\begingroup$ Thank you, I also think it is a typo! $\endgroup$
    – Bergson
    Commented Oct 12, 2020 at 2:30

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