Can I Put the Expectation Symbol Inside an Integral?

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SUMMARY

The discussion focuses on the conditions under which the expectation operator can be interchanged with an integral, specifically addressing the equation E ∫ f(x)dx = ∫ E(f(x))dx. This interchange is valid when certain conditions for the interchange of integration hold, which are rooted in the properties of the expectation operator as an integral. Key concepts include the Dominated Convergence Theorem and Fubini's Theorem, which provide the necessary framework for this operation.

PREREQUISITES
  • Understanding of the expectation operator in probability theory
  • Familiarity with integration techniques in calculus
  • Knowledge of the Dominated Convergence Theorem
  • Basic principles of Fubini's Theorem
NEXT STEPS
  • Study the Dominated Convergence Theorem in detail
  • Explore Fubini's Theorem and its applications in probability
  • Learn about the properties of expectation in probability theory
  • Investigate examples of interchanging expectation and integration
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Mathematicians, statisticians, and students of probability theory seeking to deepen their understanding of the relationship between expectation and integration.

tennishaha
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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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