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an email exchange about integral representations

integralsI had an interesting email exchange [or rather exchange of emails] with a (German) reader of Introducing Monte Carlo Methods with R in the past days, as he had difficulties with the validation of the accept-reject algorithm via the integral

\mathbb{P}(Y\in \mathcal{A},U\le f(Y)/Mg(Y)) = \int_\mathcal{A} \int_0^{f(y)/Mg(y)}\,\text{d}u\,g(y)\,\text{d}y\,,

in that it took me several iterations [as shown in the above] to realise the issue was with the notation

\int_0^a \,\text{d}u\,,

which seemed to be missing a density term or, in other words, be different from

\int_0^1 \,\mathbb{I}_{(0,a)}(u)\,\text{d}u\,,

What is surprising for me is that the integral

\int_0^a \,\text{d}u

has a clear meaning as a Riemann integral, hence should be more intuitive….

This entry was posted on April 8, 2015 at 12:15 am and is filed under Books, R, Statistics, University life with tags , , , , . You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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