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| | Tail  Probabilities: Cumulative  Distribution  Function: Asymptotic  Lower  BoundsEasyChair Preprint 15588, version history| Version | Date | Pages | Version notes | 
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 | 1 | December 18, 2024 | 5 |  |  | 2 | January 6, 2025 | 7 | (1)  Lyapunov  inequality  based  bounds  are  derived.  These  bounds  are  used  in  association  with  Mitzenmacher-Upfal  bounds (2) Application  of  the  bounds  to  reliability  theory  is  included (3) Connection  of  Chebyshev  inequality  and  Mitzenmacher-Upfal  bounds   to  Riemann  Zeta  function  is  discussed | 
Keyphrases: Chebyshev inequality, Euler Constant, Markov Inequality, Tail Probabilities, asymptotic lower bound | 
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