RDP 2015-10: The Life of Australian Banknotes Equation

L ( t ) = p ( 1 e ( e x β / k t λ ) ) = 1 2 1 e ( e x β / k t λ ) k = 1 2 p e ( e x β / k t λ ) k = 2 p 1 2 p ( e x β / k t λ ) k = ln ( 2 p 1 2 p ) e x β / k t λ = ln ( 2 p 2 p 1 ) 1 k t = λ ln ( 2 p 2 p 1 ) 1 k / e x β / k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGmb WaaeWaaeaacaWG0baacaGLOaGaayzkaaGaeyypa0JaamiCamaabmaa baGaaGymaiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTmaabmaaba GaamyzamaaCaaameqabaGabmiEayaafaGaeqOSdiMaai4laiaadUga aaWcdaWcaaqaaiaadshaaeaacqaH7oaBaaaacaGLOaGaayzkaaaaaa GccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaa aeaacaaIXaGaeyOeI0IaamyzamaaCaaaleqabaGaeyOeI0YaaeWaae aacaWGLbWaaWbaaWqabeaaceWG4bGbauaacqaHYoGycaGGVaGaam4A aaaalmaalaaabaGaamiDaaqaaiabeU7aSbaaaiaawIcacaGLPaaada ahaaadbeqaaiaadUgaaaaaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGa aGOmaiaadchaaaaabaGaamyzamaaCaaaleqabaGaeyOeI0YaaeWaae aacaWGLbWaaWbaaWqabeaaceWG4bGbauaacqaHYoGycaGGVaGaam4A aaaalmaalaaabaGaamiDaaqaaiabeU7aSbaaaiaawIcacaGLPaaada ahaaadbeqaaiaadUgaaaaaaOGaeyypa0ZaaSaaaeaacaaIYaGaamiC aiabgkHiTiaaigdaaeaacaaIYaGaamiCaaaaaeaacqGHsisldaqada qaaiaadwgadaahaaWcbeqaaiqadIhagaqbaiabek7aIjaac+cacaWG RbaaaOWaaSaaaeaacaWG0baabaGaeq4UdWgaaaGaayjkaiaawMcaam aaCaaaleqabaGaam4Aaaaakiabg2da9iGacYgacaGGUbWaaeWaaeaa daWcaaqaaiaaikdacaWGWbGaeyOeI0IaaGymaaqaaiaaikdacaWGWb aaaaGaayjkaiaawMcaaaqaaiaadwgadaahaaWcbeqaaiqadIhagaqb aiabek7aIjaac+cacaWGRbaaaOWaaSaaaeaacaWG0baabaGaeq4UdW gaaiabg2da9iGacYgacaGGUbWaaeWaaeaadaWcaaqaaiaaikdacaWG WbaabaGaaGOmaiaadchacqGHsislcaaIXaaaaaGaayjkaiaawMcaam aaCaaaleqabaWaaSaaaeaacaaIXaaabaGaam4AaaaaaaaakeaacaWG 0bGaeyypa0Jaeq4UdWMaciiBaiaac6gadaWcgaqaamaabmaabaWaaS aaaeaacaaIYaGaamiCaaqaaiaaikdacaWGWbGaeyOeI0IaaGymaaaa aiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaaGymaaqaaiaadU gaaaaaaaGcbaGaamyzamaaCaaaleqabaGabmiEayaafaGaeqOSdiMa ai4laiaadUgaaaaaaaaaaa@AE84@