Mean Lifetime

Mean lifetime in exponential decay

Derivation

• A large (but decreasing) number of elements is grouped into an assembly.
• The individual lifetime of an element is the time elapsed between some reference time and the removal of that element from the assembly.
• The total lifetime (T) is the sum of the individual lifetimes.
• The mean lifetime (τ) is the arithmetic mean of the individual lifetimes.
• The population (N) is the number of elements in the assembly at any given time.
• The initial population ($N\_0$) is the population at some initial reference time.

$N\; =\; N\_0\; e^\{-\backslash lambda\; t\}\; \backslash ,$

$-dN\; =\; N\_0\backslash lambda\; e^\{-\backslash lambda\; t\}\; dt\; \backslash ,$

$dT\; =\; t\; \backslash cdot\; -dN\; =\; t\backslash cdot\; N\_0\backslash lambda\; e^\{-\backslash lambda\; t\}\; dt\; \backslash ,$

$\backslash tau\; =\; \backslash frac\{1\}\{N\_0\}\; \backslash int\_\{0\}^\{\backslash infty\}\; dT\; =\; \backslash frac\{1\}\{N\_0\}\; \backslash int\_\{0\}^\{\backslash infty\}\; N\_0\backslash lambda\; e^\{-\backslash lambda\; t\}\backslash ,\; t\backslash ,\; dt$

$\backslash tau\; =\; \backslash lambda\; \backslash int\_\{0\}^\{\backslash infty\}\; t\backslash ,\; e^\{-\backslash lambda\; t\}\; dt$

$\backslash tau\; =\; \backslash frac\{1\}\{\backslash lambda\}$

<< Previous Word Browser Next >>
dan quisenberry world fantasy award louis alexandre berthier world fantasy convention croatian linguistic purism camcorder faverges robert munsch name decoration default argument louis charles antoine desaix de veygoux	marcel pavel montmin john franklin rank nullity theorem welsh assembly government andrew davies exponential decay front mission annabel lee red necked phalarope red phalarope	wilson's phalarope synthesis ateneo de manila university car safety brenda clough list of characters in 24 utopia planitia congenital hypothyroidism australian survivor list of celts tirthankara	siddartha znobe gramme thyroid stimulating hormone kernel (integral operator) danish minority of southern schleswig ono, hyogo ono, fukui ono, benjamin gamefaqs list of left wing publications in the united kingdom sicariidae