Is \(\phi(t) = \frac{1}{1+|t|}\) a Characteristic Function of a Random Variable?

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SUMMARY

The function \(\phi(t) = \frac{1}{1+|t|}\) is under scrutiny to determine if it qualifies as a characteristic function of a random variable. The discussion emphasizes the need to apply the inverse Fourier transform to ascertain whether the resulting function can serve as a valid probability density function. Key considerations include the properties of characteristic functions and the conditions under which the inverse Fourier transform can be computed effectively.

PREREQUISITES
  • Understanding of characteristic functions in probability theory
  • Knowledge of inverse Fourier transforms
  • Familiarity with probability density functions
  • Basic concepts of random variables
NEXT STEPS
  • Study the properties of characteristic functions in detail
  • Learn how to compute inverse Fourier transforms for various functions
  • Explore the criteria for a function to be a valid probability density function
  • Investigate examples of characteristic functions of known random variables
USEFUL FOR

Mathematicians, statisticians, and students of probability theory who are interested in the properties of characteristic functions and their applications in random variable analysis.

zibi
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Prove or disprove that function [tex]\phi(t)=\frac{1}{1+|t|}[/tex] is charcteristic function of some random variable.
 
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Okay, let's start with the definition. What is a characteristic function of a random variable?
 
you are proposing to apply inverse Fourier transform and to check whether the function we will get can be density function ? So how can inverse Fourier transform be computed in this case ? that's my question now.
 

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