Statistics of sinc function with normal argument

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SUMMARY

The discussion focuses on the statistical properties of the sinc function defined as y = sinc(x) = sin(πx)/(πx) when x follows a normal distribution N(0, s²). Key challenges include calculating the cumulative distribution function (CDF) and probability density function (PDF) due to the non-one-to-one nature of the sinc function. Participants suggest that while the CDF and PDF are complex, the moments of the distribution can be derived through numerical integration and potentially through analytical methods involving complex variables.

PREREQUISITES
  • Understanding of the sinc function and its properties
  • Familiarity with normal distributions, specifically N(0, s²)
  • Knowledge of numerical integration techniques
  • Basic concepts of complex variable methods
NEXT STEPS
  • Research numerical integration methods for calculating moments of distributions
  • Explore analytical techniques in complex variable theory
  • Study the properties of the sinc function in statistical contexts
  • Learn about the derivation of CDFs and PDFs for non-one-to-one functions
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Mathematicians, statisticians, and data scientists interested in advanced statistical properties of functions, particularly those working with the sinc function and normal distributions.

rhz
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Hi,

Can anyone point me to the pdf, cdf, moments, etc of

y = sinc(x) = sin(pi*x)/(pi*x) where x ~ N(0,s^2)?

Thanks!
 
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The CDF and PDF will be tricky to calculate because sinc is not one-to-one.

However the moments could be calculated by numerical integration and possibly even analytically using complex variable methods.
 

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