Gaussian Function Properties

This appendix collects together various facts about the fascinating Gaussian function--the classic ``bell curve'' that arises repeatedly in science and mathematics. As already seen in §B.17.1, only the Gaussian achieves the minimum time-bandwidth product among all smooth (analytic) functions.

• Gaussian Window and Transform
• Gaussians Closed under Multiplication
  • Product of Two Gaussian PDFs
  
  
• Gaussians Closed under Convolution
• Fitting a Gaussian to Data
• Infinite Flatness at Infinity
• Integral of a Complex Gaussian
  • Area Under a Real Gaussian
  
  
• Gaussian Integral with Complex Offset
• Fourier Transform of Complex Gaussian
  • Alternate Proof
  
  
• Why Gaussian?
  • Central Limit Theorem
  • Iterated Convolutions
  • Binomial Distribution
  
  
• Gaussian Probability Density Function
• Maximum Entropy Property
  • Entropy of a Probability Distribution
  • Example: Random Bit String
  • Maximum Entropy Distributions
    • Uniform Distribution
    • Exponential Distribution
    • Gaussian Distribution
  
  
• Gaussian Moments
  • Gaussian Mean
  • Gaussian Variance
  • Higher Order Moments Revisited
  • Moment Theorem
  • Gaussian Characteristic Function
  • Gaussian Central Moments
  
  
• Sums of Gaussian Random Variables