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It's easy to understand simple integrals like ∫3xdx, improper integrals, or surface integrals, but how does one interpret more complex integrals like figure 6.3 in the following images:

Complex Integral

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For example, does all of this condense down into a basic integral once the distributions/variables are known? If not, how do these integals compare with simpler ones? If someone could explain the basic parts of each integral it would be much appreciated.

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  • $\begingroup$ It might be a good courtesy to cite the source of this material if it is copyrighted. $\endgroup$ Sep 27 '13 at 22:00
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Calculating the marginal with respect to $\alpha$ of a density function with two variables $ \alpha$, $\beta$ is done by the following definition $ D'_\alpha (\alpha):=\int _{-\infty}^{+\infty} D'(\alpha,\beta)\ \mathrm{d}\beta$. This can be treated as a "normal" integral, where $\beta$ is the integration variable (usually $x$) and $\alpha$ a arbitrary parameter. So you might simply replace $\beta$ with $x$, if this makes you feel more comfortable, and solve the integrals as usual.

Marginals have less information than the original density function. You disregard the additional information of $\beta$ and consider only the density of $\alpha$. Therefore that you have to integrate over all possible values of $\beta$. Afterwards there is no dependency on $\beta$ left.

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