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What is meant by subscripting the expectation with some distribution, e.g. $\mathrm{E_{f}}[h(X)]$?

If it's any help, here's the context:

In M.C. Simulation, we wanted

$\mathrm{E[h(X)] = \int{h(x)f(x)dx}}$

so we used the law of large numbers and central limit theorem based on samples:

$\frac{1}{n} \sum_{i=1}^{n}h(X_i) \longrightarrow E_{f}[h(X)]$, with $X_{1}, \ldots, X_{n} \sim f$

tSchema
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  • I think this [answer](http://stats.stackexchange.com/questions/72613/subscript-notation-in-expectations) is clearer. – chips Mar 31 '15 at 22:19

1 Answers1

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Normally it means that you are taking the expectation with respect to that distribution (or that probability measure). Sometimes we change the probability measure and therefore the expectations are taken with respect to the new probability measure. So they want to specify exactly with respect to which probability measure, they are taking the expectations.

Stat
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    So is the thought that since $E[h(X)] = \int_{-\infty}^{\infty} h(x)f(x)dx$, it may not be clear which probability density function, $f(x)$, we're taking the expectation with respect to? So for example, if we have two pdfs, a(x) and b(x), that we potentially could take the expectation with respect to, we would write either $E_{a}[h(X)] = \int_{-\infty}^{\infty} h(x)a(x)dx$ or $E_{b}[h(X)] = \int_{-\infty}^{\infty} h(x)b(x)dx$, to make it clear which one we want to use? – tSchema Nov 24 '12 at 21:51
  • Just wanted to make sure that I understand you clearly =) – tSchema Nov 24 '12 at 21:58
  • Yes, roughly speaking ... – Stat Nov 25 '12 at 02:18