Can I Put the Expectation Symbol Inside an Integral?

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The expectation symbol can be placed inside an integral under certain conditions that allow for the interchange of the order of integration. Specifically, this is valid when the function being integrated meets criteria such as continuity and boundedness. The expectation operator functions similarly to an integral, which facilitates this interchange. It is crucial to ensure that the conditions for Fubini's theorem or the dominated convergence theorem are satisfied. Proper application of these principles ensures accurate results in probability and statistics.
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when can i put the expectation symbol inside an integral?

under what occasions does this E \int f(x)dx=\int E(f(x))dx satisfy
 
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The expectation operator is basically an integral, so you can put it under the integral sign as long as the conditions for interchanging the order of integration hold.
 

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