Expected value of the area of square

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SUMMARY

The discussion centers on the calculation of the expected value of the area of a square defined by a random variable X, where X represents the side lengths. The key conclusion is that the expected value of the area is not simply E[X]^2, as E[X]^2 does not equal E[X^2] in general cases. This distinction is crucial for understanding the relationship between random variables and their transformations, particularly in geometric contexts.

PREREQUISITES
  • Understanding of random variables and probability density functions (PDFs)
  • Knowledge of expected value calculations in probability theory
  • Familiarity with the properties of mathematical expectations, specifically E[X]^2 vs. E[X^2]
  • Basic geometric principles related to squares and area calculations
NEXT STEPS
  • Study the properties of expected values in probability theory
  • Learn about the Law of the Unconscious Statistician for transformations of random variables
  • Explore the implications of variance and covariance in random variables
  • Investigate geometric interpretations of probability distributions
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Students and professionals in mathematics, statistics, and data science who are interested in understanding the nuances of expected values and their applications in geometric contexts.

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Homework Statement


Given a square with side lengths X, where X is a random variable with some probability density function (the actual pdf is not important for my question). Why isn't the expected value of the area = E[X]^2 = E[X^2]?


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The Attempt at a Solution


Intuitively I would think, if I can find the expected value of one of the sides, I can get the expected value of the area by squaring it.
On the other hand, I am well aware that E[X]^2 != E[X^2] in the general case, and this is indeed a general case (with some geometric interpretation).
 
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