Explanation on minimization of Expected values

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SUMMARY

The minimization of the expected value E[Y - f(X)]^2 occurs when f(X) is set to E[Y|X], which is the conditional expectation of Y given X. This relationship is fundamental in statistics and ensures that the mean squared error is minimized. Additionally, the property E_y[E_x[Y|X]] = E[Y] reinforces the concept of conditional expectation and its implications in statistical analysis.

PREREQUISITES
  • Understanding of conditional expectation in probability theory
  • Familiarity with the concept of mean squared error
  • Basic knowledge of random variables and their distributions
  • Proficiency in statistical notation and terminology
NEXT STEPS
  • Study the properties of conditional expectation in depth
  • Learn about the derivation of mean squared error in regression analysis
  • Explore the implications of E[Y|X] in predictive modeling
  • Investigate the relationship between random variables and their expectations
USEFUL FOR

Statisticians, data scientists, and anyone involved in predictive modeling or statistical analysis will benefit from this discussion.

cutesteph
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Why is the E[ Y - f(x) ]^2 minimized when choosing f(x) = E[Y|X] ?
 
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Hye cutesteph.

Try finding when the expression is zero. Also since you are using two variables, what is the expectation with respect to?

Another thing to note is that E_y[E_x[Y|X]] = E[Y] (also known as conditional expectation).
 

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