2 different probability density functions will give the same E(g(x)) or not

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SUMMARY

This discussion centers on the relationship between different probability density functions (PDFs) and their expected values when applied to a function g(x). It concludes that while two distinct PDFs, p(x) and q(x), can yield the same expected value for certain functions, this is not universally applicable. Specifically, when the means of p(x) and g(x) differ, as demonstrated with the function g(x) = x!, the expected values will diverge. Thus, the expected value E(g(x)) is not invariant across all functions and PDFs.

PREREQUISITES
  • Understanding of probability density functions (PDFs)
  • Knowledge of expected value calculations
  • Familiarity with mathematical functions, specifically factorial functions
  • Basic statistics concepts, including mean and variance
NEXT STEPS
  • Research the properties of probability density functions and their implications on expected values
  • Explore the concept of expected value in the context of different mathematical functions
  • Study the impact of varying means on the expected values of functions
  • Investigate specific examples where different PDFs yield the same expected value
USEFUL FOR

Statisticians, data scientists, mathematicians, and anyone involved in probability theory or statistical analysis will benefit from this discussion.

saintman4
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If we have 2 different probability density functions p(x) and q(x). Then we find the expected values of function g(x) by using these 2 different probability density functions. Do both of them give the same expected value?
 
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If p(x) and g(x) have different means, then obviously they give different expected values for the function g(x)= x! It is certainly possible that two different density functions will give the same expected value for some functions but certainly not for all.
 

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