I understand how differential calculus is useful for basic Maximum Likelihood estimation techniques. However, my question is: what broad types of statistics require an understanding of integral calculus?
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3$\begingroup$ statistical intuition! $\endgroup$– user603Jul 8, 2013 at 19:18
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3$\begingroup$ @user603 , Can you explain what you mean? $\endgroup$– histelheimJul 8, 2013 at 19:20
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$\begingroup$ Could you please clarify what you mean by "statistics"? This word has a technical meaning (a number computed from data) but also has many common meanings covering a wide range. I'm having a hard time appreciating that this question is even appropriate here according to the site criterai which (among other things) require that "You should only ask practical, answerable questions based on actual problems that you face." $\endgroup$– whuber ♦Jul 9, 2013 at 13:59
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4 Answers
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Continuous distributions have cumulative distribution functions that involve integrals. In general
$F_X(x) = \int_{-\infty}^x f_X(x)\,dx$
where $f_X(x)$ is the PDF
answered Jul 8, 2013 at 18:53
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2$\begingroup$ and, by extension, any statistic or parameter that involves moments, correlations, etc. $\endgroup$– LucozadeJul 8, 2013 at 19:01
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$\begingroup$ +1 I doubt we'll see an answer more fundamental than that! $\endgroup$ Jul 8, 2013 at 19:08
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2$\begingroup$ This answer would be more helpful with a bit more explanation. A person (like me) who asks about the usefulness of calculus for various unknown statistical techniques is unlikely to fully understand your answer. $\endgroup$ Jul 8, 2013 at 19:13
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$\begingroup$ last line should be $f_X(x)$ instead of $f_X(t)$ (system does not allow to make changes less than 6 characters, and I did not want to spoil this beautifully succint answer!). $\endgroup$– LucozadeJul 8, 2013 at 19:26
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2$\begingroup$ @histelheim Well, you could tell us what you know in your question. Then we could give better answers. We can't read your mind or divine what your background is. $\endgroup$ Jul 8, 2013 at 19:36
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Marginalisation is a fundamental part of Bayesian statistics. Optimisation can easily lead to over-fitting and integrating over parameters, rather than optimising them, is a good way of avoiding this problem. This is an excellent motivation for acquiring as good an understanding of integral calculus as you can manage (I wish mine were much better than it is!).
answered Jul 8, 2013 at 18:45
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In addition, for discontinuous density functions, as encountered in continuous - cum - Dirac type delta mixed distributions, another type of integration (Stieltjes integrals) is still needed and useful. On the other hand, if you are looking to solve stochastic differential equations, your knowledge of standard integral calculus will be a hindrance, because you will need to learn either a new calculus (Ito calculus) or know how to deal with spurious terms in your solution (Stratonovitch calculus).
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$\begingroup$ Are stochastic differential equations related to any type of Markov models? $\endgroup$ Jul 8, 2013 at 19:08
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If finding the cumulative distribution function does not impress you in your quest for useful statistics, remember that very fundamental quantities like the mean $X$ are obtained as $\int^{+\infty}_{-\infty} x f_X(x) dx$ and the standard deviation as $\sqrt{\int^{+\infty}_{-\infty} x^2 f_X(x) dx - \left ( \int^{+\infty}_{-\infty} x f_X(x) dx\right )^2}$. Skewness and kurtosis also require similar integrations. All these tell you what their universal, i.e., population values are, for an infinite sample set. If you are not equipped with skills for integrals, you will need to stick calculating these quantities from experimental data (finite sized samples), for which you only need to do summations instead of integrations. However, these have inevitable and inherent uncertainty on their calculated values (random sampling error). So, being able to calculate such benchmark values is very useful indeed.